Dale’s A112041 Prime-Scaling Conjecture
Abstract
Four prime values at successive powers-of-three scalings force k to be four or divisible by ten.
Definition 1.1 (Dale’s divisibility conjecture).
Formalization. D5/S3/Arith/Congruence/DaleTripleScalingPrimeMultiplesOfTen.claim (✓ std3).
Citation. Pierre Cami; Harvey P. Dale (2015). OEIS A112041, Numbers k such that 1k + 1, 3k + 1, 9k + 1, 27k + 1 are all primes. URL: https://oeis.org/A112041.
Commentary.
For every positive natural k, if k+1, 3k+1, 9k+1, and 27k+1 are all prime, then k is four or divisible by ten.
Theorem 1.2 (The prime-scaling theorem).
Proof. Machine-checked in Lean as D5/S3/Arith/Congruence/DaleTripleScalingPrimeMultiplesOfTen.result (✓ std3). ∎
Resolves. Problems/oeis-a112041-dale-triple-scaling-prime-multiples-of-ten (proved) by D5/S3/Arith/Congruence/DaleTripleScalingPrimeMultiplesOfTen.result.
Source. Repository-derived.
Acknowledgement. Pierre Cami; Harvey P. Dale (2015). OEIS A112041, Numbers k such that 1k + 1, 3k + 1, 9k + 1, 27k + 1 are all primes. URL: https://oeis.org/A112041.
Commentary.
Parity first forces k to be even: an odd k would make the prime k+1 equal to two, after which 3k+1 is composite. A split into residue classes modulo five then makes one of 9k+1, 27k+1, or 3k+1 a proper multiple of five, except when k is congruent to four and the prime k+1 equals five. The remaining residue class is divisible by both two and five.