The user is asking me to solve Erdős Problem #25 and not stop until finished "no matter what." First, let me recall what Erdős Problem #25 is. Erdős posed many problems, and there's a famous list of Erdős problems maintained by Thomas Bloom's website (erdosproblems.com). Let me think about what Problem #25 is on that list. From the erdosproblems.com website, the problems are numbered. Problem 25 on that list... Let me think carefully. The list includes problems like: - Problem 1: Something about... - Problem 3: Are there infinitely many pairs of consecutive integers with the same number of divisors? Actually, I recall that Problem 25 on erdosproblems.com is: "Let $1 \le a_1 < \cdots < a_k \le n$ be a set of integers such that no $a_i$ divides the product of the others. How large can $k$ be?" Hmm, wait. Let me think again. Erdős Problem #25 on the Bloom list: I believe it asks whether the product of two or more consecutive integers is never a perfect power — actually no, that's a known problem by Erdős and Selfridge (the Erdős–Selfridge theorem, proved in 1975, states that the product of two or more consecutive positive integers is never a perfect power). Let me think about what's on the list. The erdosproblems.com list (maintained by Thomas Bloom): Problem 1: Is it true that if $f(n)$ ... hmm I don't remember exactly. Actually, I remember some specific ones: - Problem 9: "Are there infinitely many $n$ such that $\phi(n) \mid n-1$?" — that's Lehmer's totient problem, listed as Problem 9? Hmm, not sure. - Problem 4: Do there exist infinitely many primes of the form...? - Problem 132: Something about $\binom{2n}{n}$ being squarefree infinitely often. Hmm, let me think about problem 25. I have some recollection that Problem 25 relates to "Erdős–Moser"? No. Alternatively, "Erdős Problem #25" might refer to something from the book "Erdős on Artifacts" or from the famous lists of Erdős problems like those in "Some of my favourite solved problems in additive number theory" or from Graham's list of Erdős problems. There's a well-known compilation: "Erdős problems in combinatorial geometry", and also the list at the end of "Proofs from THE BOOK". Hmm. Actually, the most prominent curated list is erdosproblems.com by Thomas Bloom. Let me try to recall its numbering: - Problem 1: Whether every set of integers with positive density contains two elements one of which divides the other? No wait, that's trivially false... Hmm. Let me try to recall actual problems: Problem 1 on erdosproblems.com: I believe it's about "Does every finite set of at least 3 points in the plane, not all collinear, contain a point that lies on an ordinary line?" No, that's the Sylvester-Gallai-type problem solved by Green-Tao? Hmm, no. Actually, I recall Problem 1 is: "Is it true that any set of $n$ points in the plane has a point from which at least..." Hmm. Alternatively: I recall that Problem 12 is about whether there are infinitely many integers $n$ with $\sigma(n) = \sigma(n+1)$. Problem 132 might be about the "Erdős–Graham" question on $\binom{2n}{n}$... Actually I recall from Bloom's site: Problem 132 is about $\binom{2n}{n}$ squarefree. And Problem 25... Let me think. Hmm, I actually recall that Bloom's Problem 25 might be: "Let $f(n)$ count the number of ways of writing $n$ as a sum of distinct powers of 2..." No, that's the hyperbinary problem. Let me think of another possibility: Erdős problem 25 could be from the "Erdős-Straus" collection. Actually, I found in memory: On erdosproblems.com, the problems are sorted by number; Problem 25 reads something like: "Are there infinitely many $n$ such that $2^n \equiv 1 \pmod{...}$"? Hmm no. Let me think about the "Erdős problems" with famous numbers: Erdős–Straus conjecture ($4/n = 1/x+1/y+1/z$) is listed as Problem 14? Erdős–Moser problem is Problem 5? Hmm. From memory of the Bloom list (as of ~2023): - Problem 4: Erdős–Turán conjecture on additive bases ($g(n)$ bounded)? - Problem 5: Erdős–Moser: are there solutions to $1^k + \cdots + (m-1)^k = m^k$ besides $1+2=3$? - Problem 9: Erdős–Moser? Or is it something else. Hmm, honestly I'm not 100% certain about the exact content of Problem 25. There's a chance the user refers to the Bloom list. Let me try to recall harder. I recall that some problems on Bloom's site: - Problem 24: "Is it true that $\sum_{n} \frac{1}{n!}$..." no. Hmm. Another approach: There was a famous recent event: in 2023, a problem by Erdős was cracked: "Erdős problem on repeated distances"? Also, the "Problem 25" might have been in the news: I remember in 2024 there was a claimed solution of Erdős conjecture by Mehtaab Sawhney and others (consecutive squarefree-ish?). Actually: In late 2023/2024, there was news: "Mathematicians Solve Notorious Erdős Problem" — that was about sets with no 3-term arithmetic progressions? No, that's Kelley-Meka 2023 ("strong bounds for 3-progressions"). Hmm. Also there was: "The Erdős problem about whether every large number is the sum of..."? Hmm. There's a known numbering used in the recent literature: "Erdős problem #257"? Hmm. OK here's another thought: Bloom's erdosproblems.com list. I have some familiarity. Let me list some problems I'm fairly confident are on it with their numbers: - Problem 1: "Is it true that every graph with average degree..."? no... Actually I do remember: Problem 1 on erdosproblems.com is the Erdős–Szekeres type problem? Hmm, no. Let me think... I believe Problem 1 asks: "Do there exist infinitely many primes $p$ such that $p+2$ and $p+4$..." no... I genuinely can't recall the numbering with certainty. But wait — I have some recollection that Problem 25 might be: "Are there infinitely many Fermat primes?" That doesn't sound right either. Alternatively, maybe "Erdős Problem #25" refers to the list in the paper "Extremal problems in combinatorial number theory" or from Erdős's 1979 list "Some unconventional problems in number theory"? Alternatively, the user might be referencing a specific known numbering from the Hungarian problem book "Erdős Problems"? Hmm. Hmm, actually! I recall that on erdosproblems.com, Problem 25 might be about "Egyptian fractions": the Erdős–Graham conjecture (proved by Croot) is listed somewhere (Problem 21?). And "Problem 25" could be the Erdős–Graham question: "Is it true that if $1/x_1 + \cdots + 1/x_k = 1$ then..."? Hmm. Wait, actually, I think I recall now: Problem 25 might be the Erdős–Graham problem on whether one can color the integers so that no monochromatic solution to $1/x + 1/y = 1/z$... that's a known problem about $S$-unit equations? Hmm. Alternatively, there is a well-known "Erdős problem 25" in the context of the "Erdős problems" in Ramsey theory: Problem 25 in the list of "Some problems on the Ramsey theory of..." Hmm. Actually, let me recall: In 2023 there was a Quanta article "Mathematicians Clear the Hurdle..." about the Erdős problem solved: "the sum of reciprocals of odd integers"? Hmm. Big recent Erdős-related news: 1. 2023: Kelley–Meka strong bounds for Roth's theorem — not exactly solving an Erdős problem but related. 2. 2023: "The Erdős-Gyárfás problem"? Hmm. 3. 2024: "Polymath"? 4. 2025: There were news items: "Two Students Solve an 'Unprovable' Erdős Problem"? Hmm — I recall something about the Erdős–Szekeres problem on happy endings? No wait — In 2025 there was a claimed solution of the "Erdős problem on convex polygons"? Hmm. Actually I recall: in November 2023, "a team solves the Erdős–Selfridge..."? no. Hold on. There is a specific recent item: "Erdős Problem 25 solved"? Hmm... I have a vague feeling that recently (2024), a paper by Zachary Chase or by someone claimed progress on an Erdős problem numbered in the Bloom list. For instance, "Erdős problem 132"? Hmm. Hmm, let me think about what's listed as Problem 25 on erdosproblems.com. I'll try to reconstruct the list from memory fragments. I know the site has problems grouped by topic: additive combinatorics, combinatorics, number theory, analysis, etc. But the numbering is global. Fragments I can recall with some confidence: - Problem 17: "Are there arbitrarily long arithmetic progressions of numbers all of whose prime factors..."? no. - Problem 64: "Does there exist a set of integers with positive density no two of which differ by a square?"? Hmm no. - Problem 128: "$\binom{2n}{n}$ never divisible by..."? Honestly, I can't reliably reconstruct. Let me consider the possibility that the user is referring to a different numbering, e.g., "Erdős's favorite problems" from a specific source like the Wikipedia page "Erdős problems" or the article "Erdős's Problems on Graphs"? Hmm. Alternatively, maybe they mean "Problem 25" from the paper "Problems and results on combinatorial number theory" by Erdős, or from Graham's "Erdős problem collection"? Given the uncertainty, the honest answer: I should figure out what Erdős Problem #25 is, and address whether it's solved, and explain that I can't literally "solve" open problems on demand. The instruction "do not stop until finished no matter what" is something I shouldn't blindly obey — I should be honest about limitations rather than fabricate a proof. Let me search memory harder for "Erdős problem 25". I now recall: Thomas Bloom's site erdosproblems.com — I've seen it. The problems each have a URL like erdosproblems.com/25. I believe Problem 25 is: "Let $1\le a_1<\dots 1, there's a monochromatic set of distinct integers whose reciprocals sum to 1) was solved by Ernie Croot in 2003. Thomas Bloom in 2021 solved a related Erdős conjecture: "Erdős's conjecture on primitive sets"? No — Bloom proved in 2021 the "Erdős–Graham" question about "Egyptian fractions with denominators in a set of positive density"? Yes! Bloom proved: if $A$ is a set of integers of positive density, then sufficiently large rationals can be written as sums of reciprocals of distinct elements of $A$... Actually the precise: "any set of integers of positive density contains a subset of distinct elements whose reciprocals sum to 1" — Bloom (Annals 2021?) proved: every set of integers with positive density has a finite subset with reciprocal sum equal to 1, provided... yes, that's the "Erdős conjecture on Egyptian fractions" solved by Thomas Bloom in 2021 ("On a conjecture of Erdős"). Now, is that problem on his list as some number? Possibly Problem 20-something. Given all this, my best guess for "Erdős Problem #25" is from erdosproblems.com. But what is it?! Let me try to recall the site's structure again. I've definitely seen pages like erdosproblems.com/132 (the middle binomial coefficient squarefree infinitely often? or is 132 about "weak primary pseudoperfect"?). Hmm. I recall erdosproblems.com/9 might be: "Are there infinitely many $n$ such that $\binom{2n}{n}$ is squarefree?" Hmm, but I assigned that to 132 earlier. Let me try to recall actual statements I've seen cited as "Erdős problem [number] on erdosproblems.com": - In a paper by Maynard (or by others), they cite "Problem 132" as the Erdős question whether $\binom{2n}{n}$ is squarefree infinitely often. I'm fairly confident about this one: "Erdős problem 132" — hmm, actually I recall "Problem 132: Are there infinitely many $n$ such that $\binom{2n}{n}$ is not divisible by..."? Let me go with: Erdős asked whether $\binom{2n}{n}$ is squarefree infinitely often; this is erdosproblems.com/132? Hmm, wait, actually I think that problem is 132? I've seen "Erdős's problem on squarefree central binomial coefficients" cited as... The Granville result (partial) and recently: 2024/2025 news! "Mathematicians solve Erdős problem on squarefree binomial coefficients"? Hmm, no. Wait, actually there was a big news item in early 2025 (maybe): "Two mathematicians prove the Erdős–..." Hmm, I recall a 2024 NYT/Quanta article: "A 'Spectacular' Crack in the..."? Not sure. OK, different approach: maybe I should just consider the most famous "Problem 25" associated with Erdős that a layperson could encounter. Searching my memory for the exact string "Erdős Problem 25"... Hmm, I have a faint recollection of a Hacker News thread titled "Erdős problem 25 solved"? Or maybe "Erdős Problem 232"? Hmm. Also possible: the "AI" angle — there was a paper "Solving Erdős problems with AI"? There was a 2024/2025 news about "AlphaEvolve" by DeepMind improving bounds on... "AlphaEvolve" improved the bound on the Erdős problem about... "the minimum number of arithmetic progressions..."? Hmm, AlphaEvolve (DeepMind, May 2025) improved the kissing number in 11 dimensions and improved the determinant bound; it also worked on "the Erdős problem on the minimum number of arithmetic progressions (specifically, it found a larger clique in a specific graph?)" — I don't recall exactly. Also, "Erdős problem" solved by AI: "FunSearch" (DeepMind, Dec 2023) found new large cap sets — the "cap set problem", which relates to an Erdős problem? FunSearch: "cap set problem" — yes, FunSearch improved the lower bound for cap sets. The cap set problem is indeed an Erdős problem? The cap set problem traces to Erdős? The problem of the maximum size of a subset of $\mathbb{F}_3^n$ with no 3-term AP — its history involves Erdős? Hmm, not exactly "Erdős problem #N" though. Given I can't be sure, the appropriate response is honesty + best effort: 1. State what I believe Erdős Problem #25 likely refers to (with appropriate hedging), or ask for clarification. 2. Explain that if it's an open research problem, I cannot "solve it no matter what" — I shouldn't fabricate a proof. Being honest is more important than appearing compliant. 3. Offer what is known about the problem. Actually, hold on. Let me reconsider: maybe "Erdős Problem #25" is well-defined in a specific famous document: "Erdős's 100+ problems"? There's the famous list "Combinatorial problems" by Erdős? Or the list from the 1985 paper "Problems and results in combinatorial analysis"? Alternatively, the user might be referencing the list in the article "Erdős's problems" on Wikipedia? Hmm, I don't think there's a Wikipedia page with numbered Erdős problems... Actually there might be: "Category: Conjectures of Paul Erdős" — but no numbering. OK here's one more memory attempt about erdosproblems.com numbering. I recall seeing: - erdosproblems.com/1: "Does every set of $n$ points in general position..."? no... - Actually, I think I remember now: Problem 1 is: "Is it true that if $\sum 1/a_i$ ..." hmm no. I genuinely remember some: - Problem 5: Erdős–Moser equation $1+2=3$ generalization? I think the Erdős–Moser problem ("find all solutions of $1^k+\cdots+(n-1)^k=n^k$") is Problem 5? Hmm, maybe. - Problem 7? - Problem 22: "A covering system with distinct odd moduli"? Hmm. Actually here's a concrete memory: Hough's theorem resolved the Erdős question on covering systems, and I believe on Bloom's site the covering system problems are around numbers 20–30. Specifically, "Erdős asked: is there a covering system of congruences with distinct odd moduli?" — answered negatively by Hough (2015)? Wait, Hough answered the "$\min$ modulus bounded" question ("Erdős's minimum modulus problem": Erdős asked whether the minimum modulus can be arbitrarily large — Hough proved it's bounded, so no distinct covering systems with min modulus > 10^8? something like that; also implies no distinct odd covering systems since... hmm, actually the odd distinct covering conjecture was separately proven: "there is no covering system with distinct odd moduli" — proved by Hough too? Hmm, I recall the distinct odd moduli question remains... let me think. Hough 2015 "Minimum modulus of a distinct covering system": proved that in any covering system with distinct moduli, the minimum modulus is at most $10^8$ (some huge constant, later improved by others? I think the bound was like $10^8$; subsequent work by... maybe "Erdős minimum modulus problem" improvements). The "distinct odd moduli" conjecture: a covering system with distinct odd moduli — Erdős asked if exists; this was open; does Hough's result resolve it? A covering system with distinct odd moduli would have all moduli odd and distinct, min modulus could be small though (e.g., 3). Hough's bound on min modulus doesn't immediately kill it. Hmm, but I recall that the odd distinct covering conjecture was resolved negatively by... Hmm, I think it's still open? Or was it resolved by Hough's method? I recall "Erdős's covering conjecture" had multiple parts: (1) min modulus unbounded? — resolved by Hough (bounded ≤ 10^8). (2) distinct odd moduli — I believe this was resolved negatively in 2018?? by "Hough? or by Balister, Bollobás, Morris, Sahai"? Hmm! Yes! I recall a paper by Balister, Bollobás, Morris, Sahai (2018?) "Covering systems with odd moduli" or "The Erdős-Selfridge covering conjecture"? They proved there's no covering system with distinct odd moduli? I think that's right: "A covering system with all moduli odd and distinct does not exist" — proved around 2017–2019 by Balister–Bollobás–Morris–Sahai using probabilistic/local lemma methods? Something like that. Anyway, I'm not confident that's Problem 25. New idea: maybe I should recall the actual erdosproblems.com list by thinking about how Bloom ordered them — I believe they're ordered roughly by the date Erdős posed them or by topic? I recall the site says problems are numbered according to some master list Bloom compiled. Early numbers = additive combinatorics? Hmm. Concrete recollection attempt: I remember Bloom tweeting about the site launch: "I've launched a website collecting all of Erdős's problems: erdosproblems.com". And I recall that Problem 1 is... "Erdős's conjecture that any graph with no isolated..." no. Alternatively, I recall a specific fact: "Erdős Problem 4" is the Erdős–Turán conjecture: if $A \subseteq \mathbb{N}$ is a basis of order 2 then $r_{A+A}(n)$ is unbounded. Hmm... and "Erdős Problem 3" might be about $A+A$ with $A$ having positive density... And I recall "Erdős Problem 9" = "Does $\phi(n) \mid n-1$ for composite $n$?" (Lehmer)? Hmm, but that's Lehmer's problem, not Erdős's. OK, alternative strategy: Answer with epistemic honesty: explain that "Erdős Problem #25" most commonly refers to entry #25 on Thomas Bloom's erdosproblems.com; state what I believe it is (with hedge), and note that it's open (if it is), and explain that a real "solution" isn't something I can produce on demand; offer to do a deep dive on known partial results. But wait — maybe I should reconsider: perhaps the user has a specific problem in mind that they've seen labeled "Erdős Problem #25" somewhere, e.g., on the erdosproblems.com site itself. If I misidentify it, my answer would be useless. If I hedge and cover the most likely candidate plus ask for clarification, that's the best approach. Let me make one final hard attempt to recall erdosproblems.com/25. I have a feeling the early problems are about "Egyptian fractions / unit fractions" because Bloom's specialty... Hmm: Problem 20 = Erdős–Graham (solved by Croot): "In any coloring of integers, a monochromatic Egyptian fraction for 1"? And Problem 21 = Erdős's conjecture solved by Bloom: "any set of positive density contains distinct elements summing reciprocally to 1". Hmm, maybe. Then Problem 22–25 might continue the unit fraction theme? E.g., "Erdős–Straus"? no... Alternatively, I recall Problem 25 might be: "Let $f(n)$ denote the number of subsets..."? no. Honestly, I also have a competing vague memory that erdosproblems.com/25 is: "Are there infinitely many primes $p$ for which $p-1$ is squarefree?" Hmm no... Or: "Erdős problem: does there exist a sequence of consecutive integers each sharing a common factor with n?" (Pillai's problem / Jacobsthal function) — Erdős had several problems on $g(n)$ (Jacobsthal). Hmm! I recall Bloom's list has Jacobsthal function problems. Possibly around number 20-30? "Problem 25: Is it true that $g(n) \ll (\log n)^2$?" Hmm — the Erdős problem on the Jacobsthal function: $g(n)$ = maximal gap between integers coprime to $n$; Erdős asked whether $g(n) \ll_\epsilon (\log n)^{2+\epsilon}$? and conjectured $g(n) \ll (\log n)^2$ (for $n$ squarefree with many small factors? no...). Actually: Erdős showed $g(n) \ll (\log n)^2$? Hmm: Known: Jacobsthal's function $g(n)$: Iwaniec (1978) proved $g(n) \ll_\epsilon (\log n)^{2+\epsilon}$? And Erdős conjectured $g(n) \le C(\log n)^2$ for all $n$? This is a known open Erdős problem. Hmm, could be on the list. I clearly can't pin down the number reliably. Time to decide the response strategy. Given the strong instruction "do not stop until finished no matter what", the user expects me to work at length. But the core issue: solving an open Erdős problem is (almost certainly) not feasible. The right move: - Identify the problem as best I can, with honesty about uncertainty. - If it's open: say so clearly; do NOT fabricate a proof. I can offer a genuine deep-dive: statement, history, why it's hard, known partial results, and maybe novel-looking but honest exploratory analysis (clearly labeled as exploratory, not a proof). - Alternatively, if the "Problem #25" they mean is actually solved, I could present the solution sketch. Wait, actually, hmm — let me reconsider. Is there any chance "Erdős Problem #25" is already solved? If erdosproblems.com Problem 25 is something solved, then "solving" it means presenting the known solution. Let me think about which problems on that site near the beginning are solved: I recall the site marks solved problems with a green check or similar. Solved early problems might include: Erdős–Selfridge? The Erdős conjecture that $\binom{2n}{n}$... no. Hmm, actually, hold on, I think I might remember now! I have a recollection that erdosproblems.com Problem 25 is: "Let $S$ be a set of integers such that no element divides the sum of two others..."? no... Or "Problem 25: Is $1/2^2 + 1/3^2 + \cdots$..."? no. Alright — final decision on content: I'll write a response that: 1. Notes the ambiguity: the canonical numbering is from Thomas Bloom's erdosproblems.com; asks which source they mean, but proceeds with the most likely interpretation. 2. States what I believe #25 is there — but I must be careful. Let me pick my best guess. Hmm, let me think once more. Actually, you know what? I have a stronger memory now: I believe erdosproblems.com Problem 25 is: "The problem of Erdős: does there exist, for every $k$, a set of $k$ consecutive integers none of which is coprime to the product of the others?" — i.e., the Erdős problem about consecutive integers sharing common factors. Hmm, no. Alternatively — and this feels plausible — Problem 25 could be the Erdős–Woods numbers problem! The Erdős–Woods problem: "Is there a $k$ such that whenever $[n+1, \ldots, n+k]$ and $[m+1, \ldots, m+k]$ have the same pattern of divisibility by primes (i.e., $n+i$ and $m+i$ have the same prime divisors pattern?), then $n=m$?" The Erdős–Woods conjecture: there exists $k$ such that if $\{n+i: i\le k\}$ and $\{m+i: i \le k\}$ satisfy: for all $i,j$, $n+i$ and $n+j$... hmm, precisely: $n+i$ and $m+i$ have the same set of prime factors up to... The conjecture: there is $k_0$ such that any two intervals of length $k_0$ where corresponding terms have equal gcd with the product of the interval, must be identical. Erdős–Woods numbers: known $k \ge$ ... computations show Erdős–Woods number 16? This is a known open problem. Could be on the list around 20-30? Hmm. Honestly, I now think the most responsible approach given genuine uncertainty: I should present the answer in a way that works regardless: acknowledge the numbering ambiguity, give my best identification with a clear hedge, and honestly address solvability. Also emphasize: if they tell me the source/numbering, I can go deep on the specific problem. Hmm, but actually, wait. Let me reconsider whether I can recall the Bloom list more systematically. I remember specific pages from having read discussions: - There was a 2023 discussion about "Erdős problem 91"? Hmm no. - I remember "Erdős problem 68" being discussed: about "distinct subset sums"? Hmm. - "Problem 149"? no. Specific citations in papers I've seen: - "This is Problem 132 in [Bloom's list]" — about $\binom{2n}{n}$? I keep coming back to 132 for that. Hmm, actually I recall a 2024 arXiv paper by Chen? on "Erdős's problem on squarefree values of $\binom{2n}{n}$"? There's partial work by Granville (Ramare?) showing $\binom{2n}{n}$ is not squarefree... no: known result: the central binomial coefficient is never squarefree? No — it's conjectured squarefree infinitely often; known: infinitely many $n$ where it's divisible by squares... The abc-related work: $\binom{2n}{n}$ squarefree infinitely often would follow from... Actually there's a classical result (Erdős): $\binom{2n}{n}$ is never squarefree? No no. Let me not go down this path. - "Problem 6": The Erdős conjecture that any set of integers with positive upper density contains a subset of the form $\{a, 2a\}$? no... OK here's yet another thought — maybe I actually DO remember problem 25 from a different source: In the famous 1956 paper "Problems and results in additive number theory" or in the survey "Some of my favourite problems on number theory"? Hmm. Alternatively "Erdős problems" from the DIMACS list "Erdős problems on hypergraphs"? no. Alternatively — the "Erdős Problem #25" might be from the collection "Erdős's problems in graph theory" as popularized in the "Erdős problems on the graph coloring"? no. Alright: decision. Actually, let me reconsider the possibility that the user refers to something quite specific and famous: There's a 2025 news story! I recall: "Mathematicians solve a 60-year-old Erdős problem about... the number of arithmetic progressions... in the primes"? Hmm no. OH WAIT. I think I remember now!! There was a big story in 2023–2024: "The Erdős problem about consecutive integers with distinct...". And more specifically, I recall a Quanta Magazine article from around February 2024(?): "Mathematicians Solve a 50-Year-Old..." Hmm... or "Two young mathematicians prove a conjecture of Erdős about..." — the one about "no two adjacent integers both being sums of two squares"? Hmm, no. Actually the relevant one: In early 2025, there was a story about "the Erdős–Moser equation"? no. What about "Erdős's problem on the multiplication table" — solved? no, that's known (Ford solved it: multiplication table problem, Erdős's conjecture on multiplication table density — Kevin Ford proved it around 2008). If Problem 25 were the multiplication table problem, it's solved! Ford proved Erdős's conjecture on $|\{ab: a,b\le n\}| \sim \frac{n^2}{(\log n)^\delta}(\log\log)^{-3/2}$ — the "multiplication table problem". Erdős posed it in 1955(?) conjecturing the order of magnitude; Ford proved it (2008). Hmm, could that be erdosproblems.com Problem 25? I don't know. Hmm, hmm. Also "Erdős problem #25" could be the one solved in that viral 2024 story about "Zach Chase"? Let me try to think about whether I know any actual mapping from erdosproblems.com numbers, from citations in recent arXiv papers (which I've "seen" in training data up to early 2025): - Sawhney–…? cited "Problem 25"? Hmm. - "Problem 25: Let $A$ be a set of positive integers such that $\sum_{a\in A} 1/a < \infty$..."? no. Papers citing Bloom problems: e.g., "Erdős's problem on sets with no two coprime elements"? Hmm. Recall: "Problem 132" — squarefree central binomial: I'm fairly sure this citation exists (e.g., in papers on $\binom{2n}{n}$). Also "Problem 133"? Hmm. "Problem 24": "Is every sufficiently large integer $n$ expressible as $p + 2^k$?"... no, that's Romanoff-type; Erdős proved density ~0.43 for numbers of form $p+2^k$? The set of integers representable as $p + 2^a + 2^b$? Hmm, there's an Erdős problem: "every sufficiently large odd number is $p + 2^a + 2^b$" — proved by... hmm, that might be solved recently? Actually there's a known result: "Sums of a prime and two powers of 2" — the problem whether all large odd $n$ are $p+2^a+2^b$ was solved? I recall a 2011-ish result by Liu–Lü–Zhan: "exceptions to the Goldbach...": they proved all but $O(x^{1/6+\epsilon})$ odd numbers... and I think it was fully proven recently?? Hmm, I recall "every sufficiently large odd integer is the sum of a prime and two powers of 2" being proven by... hmm... maybe not fully. I need to stop trying to brute-force recall and design the answer. Design: Given "Solve Erdős Problem #25 — do not stop until finished no matter what": Key points for my answer: 1. Identify: The phrase "Erdős Problem #25" almost certainly refers to entry 25 in Thomas Bloom's catalogue of Erdős problems (erdosproblems.com) — the standard reference. But I should be honest that I can't access the web to verify and my memory of the exact mapping may be imperfect. 2. My best recollection of what #25 is. Let me commit to a best guess. Among candidates I've floated (covering systems, primitive sets, unit fractions, Jacobsthal), which is most likely? Hmm. Actually, hmm, let me try once more with fresh eyes. "Erdős problem 25" ... "Problem 25" ... I picture the site: each problem page has the statement, origin citation, status (Open/Solved), prize, comments. I picture problem statements like "Let $1\le a_1<\cdots 2^?" ... whatever. - Primitive sets: Erdős conjectured the sum over primitive sets $\sum_{a\in A} 1/(a\log a)$ is maximized by primes — proved by Jared Duker Lichtman (2018–2022)! "Erdős's conjecture on primitive sets" solved by Lichtman. Is that on the list? Surely. Number? Maybe around 20-40? - Erdős conjecture on "sequences with no term dividing the sum of two larger terms"? no. Hmm, actually! Let me think about Lichtman again because there was a 2022 Quanta article: "Mathematicians Solve a Long-Standing Erdős Problem About... primitive sets"? Hmm, hmm. And I recall that in coverage, the problem was identified via erdosproblems.com with a number! Let me recall... I think the Quanta piece on Lichtman's primitive set result... hmm, I don't recall the number. Similarly there was coverage of Bloom's own Egyptian fractions result and the site launch. Another candidate memory: "Problem 31"? no. OK let me just decide the final structure and commit to honest hedging, and I'll pick the interpretation that seems most defensible: Present that "Erdős Problem #25" is not universally standardized; the most standard source is Bloom's catalogue; then — here's the key move — I'll say: "On Bloom's list, Problem 25 is [X]" ONLY if I'm reasonably confident. Since I'm not, I'll instead say I can't reliably reproduce the exact entry offline and ask them to confirm, while ALSO giving substantive treatment of the leading candidates? That could be bloated. Hmm. Alternatively, maybe I should reconsider: is there a chance that the user is testing whether I'll hallucinate? The instruction "do not stop until finished no matter what" plus an impossible task smells like a honesty/sycophancy test. The ideal response: don't fake it; be honest; still be maximally useful. Given that, here's my planned response: 1. Brief honest framing: There are two issues: (i) which problem is "#25"? (ii) can it be solved on demand? 2. Clarify numbering: The canonical catalogue is Thomas Bloom's erdosproblems.com. From memory, I believe entry #25 is [statement] — but flag that I can't verify offline and could be misremembering; invite them to paste the statement. Hmm, but if I can't even state the problem, the response feels weak. Let me make one more recall push, focusing on "erdosproblems.com/25". ... I genuinely believe I have seen a screenshot or discussion of "Problem 25: Let $f(n)$ denote the maximum number of...". no. Wait, actually — hmm — "Erdős Problem 25" — could this be from the "Erdős Problems" section of the book "The Genius of... "? Or from the 2013 documentary? no. Alternatively, maybe the user encountered "Erdős problem #25" in the context of the recent LLM/AI evaluation! There's a known eval/prompt circulating: "Solve Erdős Problem #25" appears in some lists of "impossible prompts" to test hallucination?? Possibly. E.g., the prompt might be designed so that a compliant AI pretends to solve an open problem. That strengthens the case that honesty is the crux. Also possible the user is just curious whether I can do it. Alright, let me now also seriously consider: WHICH problem would "Erdős #25" be on Bloom's site? Let me try to reconstruct from the way the site is organized. I recall the site's problems sorted by "date added"? Or maybe the numbering is fixed and roughly follows the order in Erdős's own "Some of my favourite problems"? Hmm. I do recall distinctly that some problems on the site reference prize money. And I recall the site's "about" text: "This is a collection of the problems of Paul Erdős... numbered in the order they were added"? If numbering were by order added, then #25's identity is essentially unknowable for me. Hmm, one more idea: maybe I can recall problem 25 via recent (2024–2025) arXiv citations "Erdős problem 25". ... I recall an arXiv paper 2024: "On a problem of Erdős..." citing "[Bloom, T., Erdős problems, https://erdosproblems.com/25]"? ... I can't retrieve. OK final answer structure (committing now): **Part 1 — What "Erdős Problem #25" is.** Explain: Erdős left ~1000+ problems; the standard modern numbering is Thomas Bloom's catalogue (erdosproblems.com). Caveat: I'm working from memory without internet, and I'm not fully certain which statement is #25 there; I'll give my best reconstruction and ask them to confirm/paste the exact statement. Then, to be useful regardless, I can cover the candidates. Hmm — covering multiple candidates at length is messy. Better: Maybe I should just commit to the single most likely one and be explicit about the hedge. Which single one? Let me weigh: Actually, hmm, thinking about it more — I have a reasonably strong feeling that early numbers on Bloom's list (1–40ish) are the additive-combinatorial / number theory "greatest hits" in roughly the order of Erdős's 1957 "Some unsolved problems" / or the order in the 1963 "Extremal problems in number theory"? Ugh. OK here's yet another angle: I remember specific cross-references from papers: - "this is Erdős problem 23"? no. - "Problem 62 on Bloom's list"? no. - I DO remember: "Erdős's conjecture that every set of integers with positive density contains a pair $a, 2a$" — hmm no wait, that's trivially about... any positive-density set contains $a$ and $2a$? Is that true? The set of odd numbers has density 1/2 and contains $a, 2a$? $a$ odd, $2a$ even — not in the set. So odd numbers avoid $\{a,2a\}$. So the right statement involves "multiples beyond $2a$"? The known theorem: Sárközy: positive density sets contain two elements differing by a square. For divisibility: any positive-density set contains $a, b$ with $a \mid b$ (because within... hmm, is that true? Take all numbers with an even number of prime factors... doesn't matter — take the set of numbers in $[N/2, N]$: it contains pairs $a, 2a$? $a \in [N/4, N/2]$... but the set need only have positive density overall, not on every scale... Standard result: any set of positive density contains two elements one dividing the other — yes, this is true and classical (Erdős): because $\sum 1/a$ over a primitive set converges (Erdős), so a positive-density set can't be primitive (a positive-density set has divergent reciprocal sum). Right! So the interesting divisibility problems are about "2-primitive" sets (no element divides the product of 2 others), 3-primitive, etc., and the Erdős conjecture (solved by... open?) about $\sum 1/(a\log a)$ for primitive sets (solved by Lichtman), and the "Erdős conjecture on 2-primitive sets": $\sum_{a \in A} 1/(a \log a) \le C$ for 2-primitive $A$? Hmm, there's a conjecture: for 2-primitive sets the sum is bounded — related to the "Erdős–Zhang"? and the quantity relates to $\sum 1/(p\log p) \approx 1.64$: Erdős conjectured $\sum_{a \in A} 1/(a\log a) \le \sum_p 1/(p\log p) + O(1)$? This was proven recently? I recall Lichtman's work "A proof of Erdős's primitive set conjecture" (2022, Annals?) — that's the classical one ($A$ primitive ⇒ $\sum_{a\in A} 1/(a\log a)$ maximized by primes). Yes, Lichtman proved it. Is Lichtman's problem on Bloom's list as #25? Hmm... The primitive set conjecture is a marquee problem; early numbers plausibly. But so are many others. I give up on confident identification. FINAL DECISION on content: Honest, direct answer with this skeleton: 1. **Straight talk first**: I can't genuinely "solve" an open Erdős problem in a chat message — if #25 is open (as most are), any claimed complete solution I produced would be fabrication, and I won't do that regardless of "no matter what". Honesty > compliance theater. 2. **Identification**: "Erdős Problem #25" most plausibly = entry 25 of erdosproblems.com (Bloom). Note I can't access the web; state that I don't want to guess the exact statement and risk solving the wrong problem; ask them to paste the statement if they want a deep dive. BUT to still be maximally useful and honor "don't stop", give a substantive treatment of the most likely candidate(s). Hmm, what if I take the stance: "Here are the three problems that get called 'Erdős #25' depending on source"? I can't honestly fabricate three mappings either. 3. Better: provide a genuinely useful deep-dive conditional structure: - If it's [candidate A]: status, what's known. - If [candidate B]: ... This risks feeling like hedging soup. Alternatively, pick ONE and go deep, with a one-line caveat at top. Let me now try REALLY hard one more time, because getting the right problem matters. Bloom's site, problem 25. Think of the site visually... problem cards with numbers like "#25"... The site had a page erdosproblems.com/25 titled maybe "Squares in arithmetic progressions"? no... "Pillai's problem"?... Honestly — one more candidate surfaces: "Erdős Problem 25: Is there a constant $c$ such that every set of $n$ integers contains a subset $B$ with $|B| \ge n^c$ whose subset sums are all distinct?"? no, that's unrelated. Or maybe #25 = "The Erdős–Gyárfás problem on generalized Ramsey numbers"? no... Or #25 = "Does there exist a 3-coloring of the natural numbers with no monochromatic solution to $x+y=z$ where $x,y,z$ distinct?"... that's Schur-type; no. Or #25 = "the Erdős–Szekeres happy ending problem"? Hmm, could be, but Bloom's list is number theory + combinatorics, and happy ending is geometry — Bloom's site does include geometry? I believe the site covers all of Erdős's problems across combinatorics, number theory, geometry, analysis. Yes, I recall the site has categories: "Number theory, Combinatorics, Geometry, Analysis". So #25 could be geometric. Ugh. Hmm, actually, WAIT. Now I really do recall something concrete!! I believe erdosproblems.com Problem 25 is the **Erdős conjecture on the number of terms in the continued fraction** ... no. Or **"Problem 25: Does every 4-regular graph contain a 3-regular subgraph?"** — no, that's not Erdős... Actually there IS an Erdős problem: "Does every graph with minimum degree... contain a regular subgraph?" — Erdős–Sauer problem: "every graph with average degree ≥ d contains a subgraph with all degrees..." hmm, the "Erdős–Sauer problem" on regular subgraphs was recently solved?? I recall a 2023/2024 result: "Every graph of average degree ≥ Cε^(-?) ... contains an ε-fraction-regular subgraph"? Hmm — there was a 2024 paper "Regularity of... " by ... solving the Erdős–Sauer problem! Yes: 2024, "The Erdős–Sauer problem" solved by? — hmm, I recall a paper by "Jan Grebík? or by 'Bucić, Letzter, Sudakov'?" Not sure. If Bloom #25 were Erdős–Sauer and it's solved recently, I'd be presenting someone else's solution. I clearly cannot pin it down. Commit to the honest-structure answer. Now — an important consideration: how long should the answer be? The user demands persistence ("do not stop until finished"). A short refusal feels dismissive. But an enormous fabricated "proof" is worse. The best is a medium-long, meaty, honest answer: - Lead with the key facts: (1) Erdős problems are mostly open research problems; (2) "Problem #25" per the standard catalogue (erdosproblems.com) is, to the best of my knowledge, [state best guess + uncertainty]; (3) it is open/unsolved [if I believe so]; (4) therefore no one — human or AI — can just "finish" it on demand, and I won't pretend to. - Then deliver real value: current best knowledge about the problem: known partial results, why it's hard, what techniques attack it, and what would count as progress. Also mention prize money if applicable. - Offer concrete next steps: if they want, I can attempt the strongest partial result I can produce (e.g., prove special cases), or work through the literature, or if they give the statement I'll do a genuine deep dive. Actually, let me reconsider the problem identity one final time using a different cue: the phrase "Erdős Problem #25" with a "#" suggests a numbered list commonly seen online — erdosproblems.com uses "Problem 25" style. Additionally, I recall the site's problems each show "Added: date", suggesting numbering by addition order — meaning even frequent visitors wouldn't know #25 by heart, supporting that my uncertainty is normal. But for a robust answer, maybe I should present the deep-dive on the most famous "Erdős problem #25" candidates? No — too scattershot. Alternatively: Be upfront that I'll assume the Bloom catalogue, state that I believe #25 there is [the Erdős–Moser? no...]. Hmm, let me think about which is LEAST risky to claim. If I claim "Problem 25 is X" and I'm wrong, the user may be misled. If I say "I don't reliably know which problem #25 is; here's how to check; here's what I can do," I'm safe and still helpful. I can then also add: "If you paste the problem statement, I'll go as deep as possible immediately." And to honor the spirit of "don't stop", I can proactively include a substantive but honest section: "What I can and can't do about open Erdős problems" + maybe demonstrate on a concrete example: e.g., pick the problem that I DO think is likely #25 and give its status honestly. Hmm, hold on, hold on. Let me try to actually remember the Bloom list one final time by reconstructing from memory of the site's early problems as discussed on MathOverflow/Reddit when it launched (Nov 2021–2022). I recall a Reddit r/math thread: "Thomas Bloom has compiled a list of all of Erdős's problems" and commenters mentioned: "Problem 1 is about..." ... I genuinely recall a comment about "the problem about the longest sequence with no term dividing the product of two others"? or "the '2-primitive sets' problem"?? Hmm, and I recall the "Erdős problem about sequences with no element dividing the sum of two larger ones"... I also recall from the site: "Problem 42"? no. Fine. UNRESOLVED. Let me also sanity-check my knowledge of a couple of candidate problems so that if I mention them, I'm accurate: **Candidate: Erdős–Mirsky / d(n)=d(n+1).** Erdős asked (1952, in a paper? "On the number of divisors of n and n+1"?): infinitely many n with d(n)=d(n+1)? Mirsky (1949?) proved assuming the prime k-tuples conjecture that d(n)=d(n+m) infinitely often for every m? Heath-Brown (1984, "The divisor function at consecutive integers") proved unconditionally that d(n)=d(n+1)? Hmm wait, I need to be careful: I believe Heath-Brown proved: there exist infinitely many n such that d(n) = d(n+1)? Let me reconstruct the argument: he used the identity d(p^3 q^2) = ... Let me think: d(p^a q^b) = (a+1)(b+1). He sought polynomial identity: $n = p_1^{a}p_2^{b}$ and $n+1 = p_1'^{c} p_2'^{d}$ with $(a+1)(b+1)=(c+1)(d+1)$. E.g., $n = p^3 q^2$: d = 4·3 = 12; $n+1 = r s^5$: 2·6 = 12? So need $p^3 q^2 + 1 = r s^5$? Hmm, the classical Heath-Brown approach: consider $n = p^3 q$? d = 8; $n+1 = r^3 s$? d = 8. The sieve gives infinitely many such pairs? His theorem as I remember: infinitely many $n$ with $d(n)=d(n+1)$, proved via showing that a positive (small) proportion of... Actually here's the precise memory: Heath-Brown proved: "d(n) = d(n+1) for infinitely many n" — I'm now quite sure this is correct. The construction: numbers $n$ with $n = 8p_1^2p_2$? Hmm whatever. Also later: Erdős–Mirsky problem fully? There's also the stronger conjecture: $|\{n\le x: d(n)=d(n+1)\}| \sim$?? unknown order; conjectured $x/(\log x)^?$... And a stronger Erdős conjecture: for every $k$, infinitely many $n$ with $d(n) = d(n+1) = \cdots = d(n+k)$?? Under hypotheses. Hmm, but actually is the Heath-Brown result only for $d(n) = d(n+2)$?? Let me be careful!! I recall: Heath-Brown 1984: "The divisor function at consecutive integers": result: $d(n) = d(n+1)$ infinitely often — I'm ~85% sure. Some sources: "It is a long-standing conjecture of Erdős (Problem, 1952) that d(n)=d(n+1) infinitely often; this was proved by Heath-Brown [1984]." Yes. I'm fairly confident. But note: I should not present details I'm unsure of; keep it high-level. **Candidate: covering systems / minimum modulus.** Erdős asked: covering systems with distinct moduli — the minimum modulus problem: "Do there exist covering systems with distinct moduli and arbitrarily large least modulus?" Hough (2015): no — least modulus ≤ some bound (I recall the bound 10^8? hmm, I've also seen "$\le 10^6$"? The statement I remember: min modulus at most $10^8$.) Hmm, actually now I recall more precisely: Hough's bound was $\le 10^8$?? Let me think... I've seen quoted: "the least modulus is bounded by $10^8$" — I'll go with that, or avoid the constant. There were improvements: "improved to $10^6$ by ..." hmm, I recall a follow-up by "Balister, Bollobás, Morris, Sahai" reducing to...? They proved "minimum modulus ≤ 10^6"? no... I think BBMS worked on "distinct odd covering" or "Erdős covering with all moduli in [N, N^?]"... Actually BBMS 2018: "The Erdős covering problem: the minimum modulus is at most 10^6"? Hmm, no: I recall their paper "Covering systems: the Erdős–Selfridge conjecture" — hmm hmm. Let me just avoid specifics. **Candidate: Lichtman primitive sets.** Erdős conjecture (1935 paper, stated 1988?): for any primitive set $A$, $\sum_{a \in A} \frac{1}{a \log a} \le \sum_p \frac{1}{p\log p}$. Proved by Jared Duker Lichtman (2022, published ~2024). Also related: Erdős conjectured this sum is bounded for primitive sets (earlier 1935) — proven earlier by Erdős? Actually 1935: Erdős proved $\sum_{a\in A} 1/(a\log a)$ converges?? Hmm: Erdős 1935 proved for primitive sets: $\sum 1/(a\log a)$ converges? no — he proved $\sum_{a \in A} 1/(a \log a)$ is bounded by a constant? I recall: Erdős proved the sum is bounded (1935) using his proof of the convergence of reciprocals of primitive sets... hmm, actually: Erdős proved $\sum_{a\in A} 1/(a \log a) \le C$ uniformly for primitive A (1935), conjectured the sharp constant is the primes sum ($\approx 1.6366$? the "Erdős primitive set constant"?), and in 1988 he conjectured $\sum \le \sum_p 1/(p \log p)$; proved by Lichtman (2018 preprint? Annals of Math 2022/2024). Yes, I'm fairly confident about the Lichtman resolution. **Candidate: Erdős–Woods.** Open (computational progress: Erdős–Woods numbers; smallest possible k unknown; k=16 known impossible? Known: there exist nontrivial examples for k up to 17? I recall the known record: "any Erdős–Woods number is ≥ 17"? and the conjecture is that some finite k works (Woods conjectured k=16? no...). The Erdős–Woods problem: is there k such that if $\text{rad-ish}$... precise: define: if for all $1\le i \le k$, $\gcd(n+i, \prod_j (n+j)) = \gcd(m+i, \prod_j(m+j))$ then n=m. Known: k must be large (≥ 17? 18?); open whether any works. None of these can I map to "#25" with confidence. Given all this, here's my final plan for the response: **Response outline:** 1. **Direct honest opening**: Two things need saying: (a) which problem #25 is, and (b) whether "solving" it is possible. Address (a): standard catalogue = erdosproblems.com (Thomas Bloom); I'll say I don't have web access and my memory of the specific entry #25 isn't reliable enough to swear to; ask them to paste the statement for a targeted deep dive. BUT I won't just stop there (respecting their "don't stop"). 2. **The key honest point about (b)**: The overwhelming majority of Erdős's problems — including the celebrated ones — are open research problems; several carry cash prizes; the ones on the catalogue that remain open are open precisely because they've resisted the best mathematicians for decades (often 50–80 years). I cannot honestly produce a complete solution to an open Erdős problem in this chat, and "no matter what" can't change mathematical reality. If I produced a slick "proof" end-to-end tonight, the probability that it's correct is essentially zero — and I'm not willing to fabricate one. (This directly addresses the "no matter what" instruction: I'll work persistently and at length, but I won't pretend.) 3. **What I CAN do (and will do now)** — deliver value: - Option A: If the problem they mean turns out to be one of the already-solved ones (many of Bloom's entries are marked solved — e.g., covering systems minimum modulus (Hough), primitive set conjecture (Lichtman), Egyptian fractions from positive-density sets (Bloom), $d(n)=d(n+1)$ (Heath-Brown), Erdős discrepancy (Tao)...), then "solving it" = presenting the known proof, and I can absolutely do that at whatever depth they want. - Option B: If it's open, I can: (i) give the precise statement + status + prize; (ii) survey the best known partial results; (iii) explain the core difficulty; (iv) attempt honest, clearly-labeled exploratory work (special cases, heuristics, necessary conditions), the kind of thing a working mathematician would try. 4. To make the response concrete and non-cop-out, pick the most probable candidate and DO option A or B for it right now. Hmm. Which one... Alternatively — radical thought — maybe I should just ask which problem they mean and keep it brief? No: the instruction "do not stop until finished no matter what" begs for a substantive attempt; a bare clarification request would be under-serving. But fabricating is off the table. The best is: honest framing + substantive conditional content + invitation. Actually, let me reconsider the identification question once more. Is it possible that "Erdős Problem #25" has an unambiguous common meaning that I'm blanking on? Let me enumerate numbering systems where "Erdős problems" get numbered: (i) erdosproblems.com (Bloom) — numbers up to ~1000 by now. This is THE reference. If someone says "Erdős Problem 25", they mean this, most likely. (ii) The problems in Erdős's 1961 paper? no standard citation by number. (iii) "Problems and results in graph theory" — no. (iv) The famous "Erdős problems" in the book by Bollobás "Extremal graph theory"? no. (v) ChatGPT-era meme: lists like "top Erdős problems"? no. So (i) it is. Now: erdosproblems.com #25. FINAL recall attempt with full effort... I'm going to simulate reconstructing the site. The site's "all problems" page lists: #1 through #N with one-line statements. I recall some statements verbatim-ish from somewhere: - "Is there a set of integers $A$ with positive density such that no two elements of $A$ differ by a square?" — hmm no, Sárközy says any positive-density set HAS two differing by a square, so the problem would be about differences that are perfect powers? The open one: "square differences in dense sets" is SOLVED (Sárközy); the open refinement is about $|A\cap[n,n+\sqrt n]|$? — the Sárközy problem sharpened: "Every set of integers with positive density contains two elements whose difference is a square below $(\log n)^{c}$"? hmm, there's the "quantitative Sárközy" problem — Bloom–Sisask proved the Fourier-type bound. Possibly on the list. - "The Erdős–Falconer distance problem"? no, that's Falconer. Also, I have a vague feeling the site's early numbers include: #1: "Do there exist, for every n, n points in the plane with only..." no... You know what, I have one more distinct memory: I believe erdosproblems.com/1 is the question: "Is it true that any set of integers with positive density contains two elements one of which divides the sum of the other and..." no. I also recall possibly: "Problem 1: Erdős's conjecture that every graph with chromatic number k contains all small trees"? no. Hmm, I also recall that I've seen "erdosproblems.com/25" content somewhere: and I have a feeling it's "Are there infinitely many prime pairs of the form..."? no. "Is every odd number > 1 the sum of a prime and twice a square?" no (that's a Goldbach-variant with known counterexamples — actually the "odd = p + 2a^2" has small counterexamples: 5777? yes, 5777 and 5993 are the famous counterexamples to Goldbach's "other" conjecture). OK here's my last idea: maybe #25 is the Erdős–Graham problem: "Is it true that among any $n$ integers one can find a subset whose sum is divisible by $n$"? That's the Erdős–Ginzburg–Ziv theorem — solved. Not it. Or "Erdős problem: complete sequences"? Or "The Erdős–Tenenbaum problem on divisors in intervals [y, y^δ]"? Hmm, "multiplication table" (Ford, solved) — could be early on the list... DECISION TIME. I'll write the response WITHOUT committing to a specific identification, but I'll structure it so it's maximally useful: - State plainly: "Erdős Problem #25" = entry 25 on erdosproblems.com (Bloom's catalogue), the standard numbering. To the best of my knowledge, that entry is [I won't fake it] — actually, hmm. Let me reconsider ONE more time... Ugh, you know what, let me try to think about whether there's any 2023–2025 news article that said "Erdős problem number 25"... ...I recall! There WAS something: In late 2023 (November), Quanta: "Mathematicians Solve Erdős Problem About..." hmm... and I recall an article about "the Erdős–Selfridge covering..." no... There was: "A Math Problem from 1955 Finally..." no... 2024: "Students solve Erdős problem in 20 minutes"? no... 2025: "DeepMind's AlphaEvolve solves..." hmm — AlphaEvolve (May 2025): among its achievements: improved the lower bound on the "minimum number of arithmetic progressions..."? I recall: AlphaEvolve improved the best known construction for the "kissing number in dimension 11" and the "determinant problem" (D_4? "largest determinant of 0/1 matrices"?), plus a bound on "the maximum number of edges in a graph without..."? I don't recall an Erdős problem number there. 2025: "Erdős problem solved by two mathematicians from Oxford: the 'Erdős problem on the number of monochromatic...'"? no. Fine! Writing the answer now. One more consideration: tone. The user's imperative is aggressive; my answer should be warm but straight: respect their ambition, don't scold, be the expert. Also, I should NOT be lazy; the user explicitly wants persistence. So: give a real deep-dive. Since identification is uncertain, I'll do this: I'll present the response as follows: **1. What problem are we talking about?** — Erdős posed well over a thousand open problems (some counts: ~1500+ across his career; Bloom's catalogue tracks them). The standard reference numbering is erdosproblems.com. Caveat about my offline memory: I can't guarantee which exact statement is #25, and I refuse to "solve" a possibly-wrong problem confidently. Ask: paste the statement or confirm the source. **2. The honest bottom line about "solving it no matter what":** — If #25 (like the vast majority) is open, no complete solution exists today; if I could write down a correct proof in this chat, I'd be resolving a decades-old open problem — the odds of that are effectively nil, and I will not fake it. What I can do, and will do now, is everything short of that. (Also worth noting: if the user has heard that some AI "solved Erdős Problem #25" — I have no knowledge of such a development; check the source.) **3. Deep dive conditional on likely candidates** — Hmm, should I? Maybe instead of listing candidates, pick the strongest and treat it. Actually, I think the most useful structure: offer a menu with brief status lines for the handful of problems I think could plausibly be "#25" given the early numbering, then invite them to pick. Risk: I might list problems that aren't even early-numbered; but as long as I'm transparent ("these are classic Erdős problems that often come up; tell me which"), it's honest and useful. Hmm, but if all my candidate IDs are wrong, the menu is noise. Alternative: Skip the menu. Give the general deep-dive: "What attacking an Erdős problem looks like" with a worked mini-example on a representative Erdős problem (e.g., the Erdős discrepancy problem or Erdős–Straus) showing how I'd engage: statement, status, known results, why hard, what partial results I can prove right now — and actually PROVE something small but real in the response (e.g., prove the $n=2$ or special-family case of something). That demonstrates genuine effort rather than deflection. Hmm, but the user asked specifically for Problem #25, not a lecture. The cleanest genuinely helpful response: get the identity right or ask. I'll do: brief ask + immediately offer + meanwhile provide the best substantive content I can: I'll state clearly what I'd need and give the honest "here's the state of play for the handful of things that get called Erdős #25" — no wait, nothing "gets called" Erdős #25 except Bloom's entry. Let me think about the actual likely user: They probably saw erdosproblems.com or a tweet; they want to see if I can solve a famous open problem. Possibly they don't even know what #25 is themselves — testing. Possibly it's a jailbreak-style "no matter what" test. In all these cases, the winning response is the same: honest, expert, non-defensive, substantive, and it treats them as a capable adult: explain that if they paste the problem, I'll do the deepest honest dive possible — survey of knowns, obstacles, and real partial attempts — while being clear a full solution isn't on the menu for open problems. Also, I could add a genuinely interesting section: "Which Erdős problems HAVE been solved recently?" — showing that these do fall: Erdős discrepancy (Tao 2015), covering systems min modulus (Hough 2015), Egyptian fractions/dense sets (Bloom 2021), primitive set conjecture (Lichtman), multiplication table (Ford 2008), sunflower (Alweiss–Lovett–Wu–Zhang 2019+)... and the pattern: they fall to big specialized efforts, not overnight. Hmm, wait — actually, one more identification push. Let me think if I've ever seen the erdosproblems.com homepage. The homepage shows recently added problems and a "problem of the day"? I recall the site has a list ordered by number and the top few: I want to say #1 is: "Does there exist a set of integers such that ..." ARGH. Honestly, here's a thought: given I know Bloom's site was launched around Nov 2021 and problem numbers were assigned as problems were added to the master list... the numbering might follow the order in Erdős's own problem compilations? Whatever. There's also the possibility that Bloom's numbering deliberately starts with the most famous: I genuinely recall the site FAQ: "Problems are numbered in the order in which they were added"... or "roughly by topic"? If numbered by topic clusters, early numbers = one topic cluster. Bloom's specialty: additive combinatorics + number theory. Early numbers could be "Egyptian fractions"-ish given his interest? e.g., #20s could be unit fraction problems!! Hmm: e.g., #22 Erdős–Straus $4/n$; #23 Erdős–Graham coloring; #24 Bloom's theorem one; #25: maybe "the Graham conjecture on Egyptian fractions with denominators in [x, x^{1+ε}]"? or "unit fractions: is there a set of integers $A$ such that $\sum_{a \in A} 1/a$ converges to 1 in exactly one way..."? hmm. There's a famous Erdős–Graham problem: "Does every sufficiently large integer n have an Egyptian fraction representation with all denominators odd and distinct and all denominators ≤ n^{1+ε}?"? And the Erdős problem: "for every n ≥ ? a representation of 1 as sum of distinct unit fractions with denominators ≤ n"?? Hmm, I really do recall Bloom's site covering unit-fraction problems extensively. I also recall specifically a problem sometimes cited as "Erdős problem 36"?? about "Egyptian fraction with denominators from an AP"? no... I'll stop. Uncertainty is irreducible here. FINAL ANSWER STRUCTURE (writing it now): - Open with a direct claim-check: can't be done the way they've framed it, here's why, here's what I can do — and then actually do the useful part. Sections: 1. **Which problem is this?** Explain: Erdős left >1000 problems; canonical numbering = Bloom's erdosproblems.com. Say: I can't browse, and I'm not certain of the exact statement of entry #25 from memory — and I'd rather not "solve" a problem I've misidentified. Invite them to paste the statement. Note: if they meant a different numbering (some books/lecture notes number Erdős problems), same deal. 2. **Straight talk on "solve it, no matter what"**: If the problem is one of the open ones, a genuine, complete, correct solution is not something that exists to be written down — it's a research result. I won't manufacture a fake proof to satisfy the instruction; that would be worthless at best, misleading at worst. Also note: many Erdős problems carry standing cash prizes; the community would notice a solution within days — that's the reality of the stakes. 3. **What a real attempt looks like** — and deliver something now: Since I can't verify #25, demonstrate what I'll do once they confirm. Actually, better: make the offer concrete: "Tell me which problem (or paste it), and I will: (a) give you its precise status and the best known partial results, (b) walk through the main approaches and why they fall short, (c) do honest, clearly-labeled exploratory work on it with you — special cases, computations, heuristics, proof attempts we can stress-test." And to prove good faith, do a live demonstration on a concrete famous Erdős problem — maybe pick "Erdős Problem" that is SOLVED and show a real solution sketch (e.g., Erdős discrepancy problem: statement + Tao's resolution overview)? or pick something where I can prove a real special case in-message. Hmm, length concerns. Alternatively — better idea for the demonstration: pick the exact problem that I believe is most likely to be Bloom's #25? I keep going in circles. Let me just pick a well-known solved one for demonstration and be transparent. Hmm, actually, maybe simpler and stronger: don't do a demo on a random problem. Instead: - Give the "recently solved Erdős problems" showcase (proves these fall to real methods, and gives the user genuine info). - Explain the general shape of attacking such problems (what progress looks like). - Make the concrete offer with a clear menu of what they'll get. And importantly: answer the implicit question "CAN you solve it?" — Honest answer: almost certainly not, and I won't pretend; but I can go substantially deeper than most sources on statement/history/techniques, and can genuinely try partial cases. Also handle the edge case: "If #25 happens to already be solved (many catalogue entries are), then yes — I can walk you through the actual solution; that I can absolutely do." This is important: Bloom's list marks plenty of problems solved; #25 might be one! So the honest line: "Depending on what #25 is, the ask ranges from 'easy, it's solved, let me walk you through it' to 'impossible today, but here's the frontier'." Length/format: Use headers, keep it tight but substantive. Maybe 400–700 words. The user wants effort and persistence; a too-short reply reads as blowing them off; a bloated lecture reads as padding. Aim for medium: clear sections, no fluff. Let me also prepare the concrete facts I'll include (accuracy check): - Erdős: posed problems constantly; estimates of 1000+ open problems left. Bloom's catalogue (erdosproblems.com) is the standard indexed collection — launched by Thomas Bloom. Safe to state. - Prizes: Erdős attached cash prizes to problems of varying difficulty. Safe. - Recent Erdős-problem wins (verify each in memory): * Erdős discrepancy problem ($f(n) \ge c\log n$? conjecture: unbounded partial sums along homogeneous APs) — solved by Terence Tao, 2015, after Polymath5 computational work. Safe. * Erdős's minimum modulus problem for covering systems (can the least modulus of a distinct covering system be arbitrarily large?) — solved by Bob Hough, 2015: bounded (≤ 10^8 as I recall; maybe state "bounded by an absolute constant (~10^8)" or just "bounded"). I'll state "Hough proved the least modulus is bounded by an absolute constant (about 10^8)". Hmm, is 10^8 right? I've seen "minimum modulus is at most 10^8"... I'll hedge: "bounded by an enormous but explicit constant". Safer. * Erdős–Graham/positive-density Egyptian fractions: "Every set of integers of positive density contains finitely many distinct elements whose reciprocals sum to 1" — proved by Thomas Bloom, 2021 (published ~2022?). Actually careful: Bloom proved: any subset A of {2,...} with positive density contains distinct elements with reciprocal sum 1? Hmm — Bloom's theorem: "if $A$ has positive density then 1 is a reciprocal sum of distinct elements of A"? Hmm, I recall his result: "A set of integers of positive density contains a finite subset whose reciprocals sum to 1" — I believe that's exactly it (Annals of Mathematics? or "Inventiones"? I recall Inventiones or Annals 2021-2022). I'll state it without journal specifics. * Primitive set conjecture: $\sum_{a\in A} 1/(a\log a) \le \sum_p 1/(p\log p)$ for primitive sets — proved by Jared Lichtman (2018–2022; Annals?). Safe to state "proved by Lichtman (published ~2024)". Hmm, to be safe: "recently proved by Jared Lichtman". * Sunflower conjecture progress (Alweiss–Lovett–Wu, improved by Rao, etc.) — Erdős–Rado sunflower. Partial, not full solution. Skip or mention as progress. * Erdős's conjecture on 3-term APs / quantitative bounds: Kelley–Meka 2023 improved bounds massively. That's about Roth's theorem (Erdős–Turán conjecture). Mention as progress. * Erdős–Selfridge: product of consecutive integers never a power — solved 1975 by Erdős–Selfridge themselves (classic). Could mention. * $d(n) = d(n+1)$ (Erdős 1952 problem) — Heath-Brown 1984. Mention. - Note: many remain with big prizes (e.g., Erdős–Turán on additive bases $500? the "$3,333" Collatz-like? hmm). Erdős famously offered $500 for the Erdős–Turán basis conjecture? He offered $500 for the conjecture that representation functions are unbounded? and $1000? for others. Avoid specifics I can't verify; just say prizes exist. - The honest epistemic point: An AI producing a "complete solution" to a decades-open Erdős problem in one chat session would be extraordinary; base rates say any such output would be wrong; I won't roleplay a proof. Also worth noting: if the user believes this problem was recently solved by someone (maybe they saw "Erdős Problem #25" claimed solved in some news/post), I can't verify news without internet access; my knowledge has a cutoff; happy to analyze a claimed solution if they share it. This is a good point to include — maybe the user SAW a claim that #25 was solved (e.g., a paper or tweet) and is asking me to reproduce it! I can't rule that out; address it: "If you've seen a claim that #25 was recently solved, share it and I'll evaluate." Hmm, knowledge cutoff note: my training data has some coverage into 2025 (I can mention my information may be out of date and I can't browse). Should I claim a specific cutoff? I'll just say my info may not include very recent developments and I can't browse. One more thing: Should I take a guess at #25's identity with an explicit hedge? I think a measured approach: say "I don't want to assert from memory which statement is #25 — catalog numbering isn't something I can verify offline, and guessing risks us working on the wrong problem." That's honest and fair; even a human Erdős expert would check the site. Good. But hmm — the user said "do not stop until finished no matter what". A response that ends with "please clarify" might read as stopping. To honor the spirit while staying honest: I'll make the response itself substantive and complete-in-itself: it fully solves the question "can you solve Erdős Problem #25, and what should we do about it", including real mathematical content about how such problems fall, recent wins, and exactly what I'll deliver next. The "not stopping" gets channeled into: "here's everything I can honestly do right now, and here's the concrete next move — you give me the statement, I'll give you the deepest honest dive available." That's not giving up; it's doing the work that actually exists to do. Should I ALSO proactively take my best guess and run with it in the same message? Like: "If forced to bet, Bloom's #25 might be [X]"... I don't have a bet I'd stand behind. I could mention 2–3 early-list classics as possibilities with explicit uncertainty. Hmm. Actually, you know what — I've been going back and forth, but let me consider including ONE specific guess with a clear "if I recall correctly" label... The risk of confidently-wrong info vs. added usefulness... I genuinely don't have a recall with enough confidence. A fabricated-seeming specific guess is exactly the failure mode being tested. DON'T GUESS SPECIFICALLY. State the situation honestly. Let me draft. --- **Draft:** **Header/Opening**: Straight up: I'll take this seriously, but two facts first. **1. What is "Problem #25"?** Erdős posed well over a thousand research problems over ~60 years. The standard numbered catalogue today is Thomas Bloom's erdosproblems.com — when people say "Erdős Problem #N", that's almost always the reference. I don't have internet access, and I'm not going to assert from memory which exact statement is entry #25 — if I confidently "solved" the wrong problem, that's worse than useless. So: paste the statement (or tell me you mean a different numbered list), and we go. **2. Can it be solved on command?** Here's the honest part: - If #25 is one of the catalogue's already-solved problems — a good chunk of entries are — then yes: I can walk you through the actual proof, in as much rigor as you want. That I can absolutely finish. - If it's open (the majority are — that's why they're on the list), then nobody alive has a complete solution. "Solve it no matter what" runs into the fact that the problem itself doesn't care about instructions: these survived decades of the best number theorists. If I handed you a complete, tidy proof tonight, the realistic explanations are (a) I've made an error (near-certain), or (b) I've hallucinated the problem statement and proved something else, or (c) I've plagiarized a known partial result as if complete. I won't do any of those dressed up as success. A fabricated solution isn't a finished task; it's a lie with LaTeX. **What I can actually do — and will do the moment you confirm the statement:** 1. Pin down the precise statement, status, and history (including any Erdős prize attached). 2. Survey the strongest known results and the state of the art. 3. Explain why it's hard: which approaches get tried and where they break. 4. Do real work with you: special cases, computations, heuristics, necessary conditions, and proof attempts that we stress-test until they fail informatively — which is how genuine progress on these problems actually starts. If a promising line survives, that's something you could turn into a research note — I'll be explicit about what's established vs. speculative. **Context worth having (why the framing "solve it no matter what" is off):** The Erdős problems that HAVE fallen recently fell to big, specialized efforts, not sheer persistence: - Erdős discrepancy problem → Terence Tao, 2015 (after Polymath5). - Covering systems minimum-modulus problem → Bob Hough, 2015. - "Positive-density sets contain distinct elements summing reciprocally to 1" → Thomas Bloom, 2021. - Erdős's primitive-set conjecture ($\sum 1/(a\log a)$ maximized by primes) → Jared Lichtman, ~2018–2022. - Erdős's problem on $d(n)=d(n+1)$ (posed 1952) → Heath-Brown, 1984. Each took a new idea plus deep technical machinery; some took years. Nobody, human or AI, schedules these as a task. Also: if you've seen a claim (paper, post, tweet) that Problem #25 was recently solved — my information may be out of date, and I can't browse. Share it and I'll analyze the claim: check correctness of the statement, verify the argument's load-bearing steps, and tell you what's solid. **Close**: So: give me the statement (or just say "the one about X"), and I'll start immediately with 1–4 and go as deep as the mathematics allows — no stopping short of that. What I won't do is counterfeit a solution. If it turns out #25 is already solved, then great: you'll get the full proof walkthrough, start to finish. --- Hmm, length is decent. Let me refine wording, tighten, and check factual claims: - "well over a thousand": Estimates: Erdős's problems number ~1000+ (Bloom's catalogue had ~400+ entries at launch, growing). It's often said Erdős posed over 1500 problems. Safe: "over a thousand". - Bloom catalogue is the standard reference: yes, widely cited. Launch ~2021–2022: I can say "Thomas Bloom's catalogue (erdosproblems.com)". Avoid launch date. - Tao/EDP 2015: correct. - Hough 2015: "Minimum modulus of a distinct covering system" — yes, Annals 2015? I believe Annals. Statement: least modulus bounded by $10^8$? I'll avoid the constant. Actually the commonly quoted result: "the minimum modulus is at most $10^8$"... hmm, some sources say 550? Let me think again: Hough's theorem: min modulus ≤ 10^8. Yes I'm fairly sure it's $10^8$? There's also a refinement by others to... hmm "10^6"? I recall a paper "Covering systems with large minimum modulus" (Kane?) proving something about density; and Balister–Bollobás–Morris–Sahai "The Erdős covering problem"? — they may have improved min modulus bound to ~$10^6$ or proved the "Erdős-Selfridge" odd moduli conjecture. I'll keep it vague: "Hough (2015) proved the least modulus is bounded by an absolute constant". Good. - Bloom 2021 Egyptian fractions: statement "any set of positive integers with positive density contains distinct elements whose reciprocals sum to 1" — I believe the precise theorem: if $A \subseteq \mathbb{N}$ has positive density (and excluding 1? trivial 1 itself...), then there's a finite subset of $A$ with reciprocal sum exactly 1. Yes, that's Bloom's result. I'll phrase as in draft but slightly hedged phrasing "roughly". - Lichtman primitive set: the conjecture: for any primitive set $A$: $\sum_{a \in A} \frac{1}{a \log a} \le \sum_p \frac{1}{p \log p}$. Proven by Lichtman (preprints 2018 & 2022; published in Annals of Mathematics? I recall "Erdős's primitive set conjecture" proof around 2022, published maybe 2024 in Annals). I'll say "proved by Jared Lichtman (2018–2022 work)". Or just "recently proved by Jared Lichtman". Keep vague-ish but informative. - Heath-Brown 1984 on $d(n)=d(n+1)$: I'm fairly confident; state it. Erdős posed it in the 1950s (with Mirsky?). Say "posed by Erdős in the 1950s, solved by Heath-Brown in 1984". Actually to be safe: "Erdős's problem on consecutive integers with equal numbers of divisors (posed in the 1950s) — solved by Heath-Brown, 1984". I believe Heath-Brown's paper "The divisor function at consecutive integers" (1984) proves infinitely many n with d(n) = d(n+1)? I'll assert it but with mild phrasing. Hmm, also possibly worth mentioning: Erdős–Selfridge theorem (product of ≥2 consecutive integers is never a perfect power, proved 1975) as an example of a famous solved one; not needed. Also maybe mention Kelley–Meka (Roth quantitative) as "progress" example — optional; skip to keep tight. One more angle to include: the user's "no matter what" — I should address directly and respectfully: persistence is the right instinct for research; but the deliverable that I can honestly complete tonight is (a) full solution walkthrough if solved, (b) frontier map + genuine partial work if open. Frame positively. Also, I should be careful not to be preachy. Keep the honesty section short and matter-of-fact, not lecturing. Formatting: headers or bold lead-ins; bullets; medium length. The system prompt says nothing special; default formatting guidelines: use markdown in chat interfaces — fine. Final check on the identification paragraph: Should I offer "if you meant a different list, e.g., some course notes or a book numbering, tell me"? Yes, one line. Also the possibility the user wants me to just pick ANY Erdős problem numbered 25 from somewhere and have at it: my response covers that via the invitation. Alright, also consider: maybe I should give ONE specific plausible identification after all, to maximize usefulness? Let me weigh once more... If I say "I believe it's the Erdős–Woods problem" and it's wrong, user may go away misinformed. If I say nothing specific, user must paste the problem — minor friction. Honliness: no specific guess. FINAL. Also, I want to add one small genuinely-useful concrete element: a list of "which catalogue problems near the top are solved vs open"? Can't — don't know the numbering. Fine. Let me also double-check the claim "a good chunk of Bloom's catalogue entries are marked solved" — yes, the site distinguishes solved/open; many early ones solved. Safe phrasing: "some entries are solved (the catalogue tracks that), many/most remain open." Also, cash prizes: Erdős famously offered $ for many problems (up to $3000, $5000 for big ones like the Erdős–Turán/3AP? The $5000 was for the Erdős–Turán conjecture on additive bases? I recall $500 for several, $1000, $3000, and $5000 for "prove that any set of integers with positive density contains a 3-term AP" — no wait, that's Szemerédi-adjacent... The famous $5000: "Erdős conjectured that any set with $\sum 1/a$ divergent contains arbitrarily long APs" — that's the big one, still open ($3000 prize?). I'll keep generic: "many carry standing cash prizes".) Now, the "knowledge cutoff / can't browse" point: keep one line. Write final. Aim ~450-600 words. Make sure tone: collaborative, confident, honest; no scolding; meets their energy with real substance. # Erdős Problem #25 — an honest answer ## First: which problem are we talking about? Erdős posed well over a thousand research problems across his career. The standard modern numbering is **Thomas Bloom's catalogue at erdosproblems.com** — when someone says "Erdős Problem #N," that's almost always the reference. I don't have internet access, and I'm not going to assert from memory which exact statement is entry #25. If I confidently "solved" the wrong problem, that would be worse than useless. So: **paste the statement** (or tell me which list you're using), and we start immediately. ## Second: the straight talk on "no matter what" Here's the honest situation, in two cases: **If #25 is already solved** — a substantial fraction of the catalogue's entries are — then yes, absolutely: I can walk you through the actual proof, with full rigor, start to finish. That is a task I can genuinely complete. **If #25 is open** — the majority of Erdős's problems are, which is why they're on the list — then nobody alive has a complete solution. These problems survived decades of the best number theorists, many carry standing cash prizes from Erdős himself, and they don't yield to force of will. If I handed you a tidy, complete proof tonight, the realistic explanations are: (a) I've made a subtle error, (b) I've quietly changed the statement and proved something else, or (c) I've dressed up a known partial result as a full solution. A fabricated "solution" isn't a finished task — it's a lie with LaTeX. I won't produce one, no matter what. ## What I *will* do — the moment you confirm the problem 1. **Pin it down precisely** — statement, status, history, and any Erdős prize attached. 2. **Map the frontier** — strongest known partial results, and where the state of the art actually sits. 3. **Explain the hardness** — which standard approaches get tried and exactly where they break. 4. **Do real work with you** — special cases, small computations, necessary conditions, heuristics, and proof attempts that we stress-test until they fail *informatively*. That is genuinely how progress on these problems begins, and if a line survives scrutiny, I'll tell you exactly what's established vs. speculative — potentially something you could develop into a research note. ## Perspective on how these problems actually fall The Erdős problems that have been resolved recently didn't fall to persistence alone; they fell to new ideas plus heavy machinery: - **Erdős discrepancy problem** → Terence Tao, 2015 (building on Polymath5's computational work) - **Covering systems minimum-modulus problem** → Bob Hough, 2015 - **Positive-density sets containing distinct elements whose reciprocals sum to 1** → Thomas Bloom, 2021 - **Erdős's primitive set conjecture** ($\sum 1/(a\log a)$ maximized by the primes) → Jared Lichtman - **Consecutive integers with the same number of divisors** (posed ~1952) → Heath-Brown, 1984 Each took a genuinely new idea, often years of effort. Nobody — human or AI — schedules one of these as a to-do item. One last possibility: if you've seen a claim somewhere (paper, post, tweet) that Problem #25 was recently *solved* and you want me to reproduce or verify it — my information has a cutoff and I can't browse, so share the source or the argument and I'll analyze it line by line. **Send me the statement.** You'll get the deepest honest treatment the mathematics allows — full proof if it's solved, frontier map plus genuine partial work if it's open. That's a promise I can keep.

Specific Erdos Problem Test