Reused writeup on Math-6128 (gpt pro 5.2)

Erdős Problem 1074 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/1074

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/1074.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/1074.tex.

1) FORMAL RESTATEMENT Define the set S of all integers m >= 1 such that there exists a prime p with two properties: (a) p divides $m!+1$, and (b) p is NOT congruent to 1 modulo m, i.e. $p-1$ is not divisible by m.

Define the set P of primes p such that there exists an integer m with the same two properties: p divides $m!+1$ and $p-1$ is not divisible by m.

Questions: - Does the asymptotic density of S exist? - Does the asymptotic density of P among the primes exist?

2) QUICK LITERATURE/CONTEXT CHECK From the problem statement: - Erdős, Hardy and Subbarao proved that S and P are infinite. - The first values of S are: 8, 9, 13, 14, 15, 16, 17, ... - Hardy and Subbarao checked up to m <= 2^10 and found m in S for m >= 8. No other external results are assumed here.

3) ATTACK PLAN To study S and P one needs structural constraints on prime divisors of m!+1 (especially congruence information), and at least small exact computations to confirm the initial segment.

4) WORK Lemma 1074.1 (Any prime divisor of m!+1 exceeds m). If p is prime and p divides $m!+1$, then p > m. Proof. If p <= m then p divides m!, hence p divides $m!+1 - m!$ = 1, impossible. Therefore p>m. QED.

Lemma 1074.2 (Wilson complement identity specialized to a prime divisor of m!+1). Let p be an odd prime and suppose p divides $m!+1$ for some m with 1 <= m <= p-2. Then $p-1-m$! ≡ $-1$^m (mod p). In particular, if m is odd then p divides $(p-1-m)! + 1$. Proof. Apply Lemma 1072.3 with n=m: for any odd prime p and 0<=m<=p-1, m! * $p-1-m$! ≡ $-1$^$m+1$ (mod p). If additionally p | $m!+1$, then m! ≡ -1 (mod p). Substituting gives $-1$ * $p-1-m$! ≡ $-1$^$m+1$ (mod p), hence $p-1-m$! ≡ $-1$^m (mod p). If m is odd then $-1$^m = -1, so p divides $ (p-1-m)! + 1 $. QED.

5) VERIFICATION (small cases) For m <= 20, m!+1 is small enough to factor exactly, so membership in S can be checked exactly. The computed S intersect {1,...,20} equals: S cap [1,20] = {8, 9, 13, 14, 15, 16, 17, 18, 19, 20}. Witness primes p (one example each) are: m=8 : p=61 divides 8!+1 and $61-1$ mod 8 = 4 m=9 : p=71 divides 9!+1 and $71-1$ mod 9 = 7 m=13 : p=83 divides 13!+1 and $83-1$ mod 13 = 4 m=14 : p=23 divides 14!+1 and $23-1$ mod 14 = 8 m=15 : p=59 divides 15!+1 and $59-1$ mod 15 = 13 m=16 : p=61 divides 16!+1 and $61-1$ mod 16 = 12 m=17 : p=661 divides 17!+1 and $661-1$ mod 17 = 14 m=18 : p=23 divides 18!+1 and $23-1$ mod 18 = 4 m=19 : p=71 divides 19!+1 and $71-1$ mod 19 = 13 m=20 : p=20639383 divides 20!+1 and $p-1$ mod 20 != 0. From these m<=20 computations, the set of Pillai primes P encountered is: {23, 29, 59, 61, 67, 71, 83, 137, 139, 269, 479, 661, 537913, 1000357, 1059511, 20639383, 75024347, 3790360487, 117876683047, 1713311273363831}.

6) FINAL UNRESOLVED (i) Strongest proved partial result here: any prime divisor p of m!+1 must satisfy p>m (Lemma 1074.1), and such a divisor forces a complementary factorial congruence (Lemma 1074.2). Exact computation confirms the stated initial segment S cap [1,20] = {8,9,13,14,15,16,17,18,19,20}. (ii) First gap (crisp): Determine whether the limit density of S exists (and similarly for P among primes). Even proving that S has density 1 (or density 0, or does not converge) is not resolved here. (iii) Top 3 next moves (concrete): 1. Prove a criterion ensuring m in S that can be checked without fully factoring m!+1, e.g. by guaranteeing a prime divisor p of m!+1 with $p-1$ not divisible by m. 2. Establish upper and lower bounds for |S cap [1,x]| that are within o(x) of each other (or exhibit oscillations). 3. Compute S for much larger m with rigorous methods (e.g. finding at least one certified prime divisor p of m!+1 and checking $p-1$ mod m). (iv) Minimal counterexample structure: If S failed to have density 1, one would need infinitely many m for which every prime divisor p of m!+1 satisfies $p-1$ divisible by m.

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.