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Kagey’s Power-Product Conjecture Is False

Abstract

The exponent coordinates of 12 and 72 refute eventual equality in A343881.

Definition 1.1 (A power-product equation).

Formalization. D5/S3/Arith/Factorization/OeisA343881KageyPowerProducts.Works (✓ std3).

Citation. P. Kagey; OEIS Foundation Inc. (2021). OEIS A343881, power-product minima and A064549. URL: https://oeis.org/A343881.

Commentary.

Table read by antidiagonals upward: T(n,k) is the least integer m > k such that k^x * m^y = c^n for some positive integers c, x, and y where x < n and y < n; n >= 2, k >= 1. (OEIS A343881, NAME, revision 16; unpaginated.) All variables are natural numbers. Positivity of c, x and y and the strict bounds on both exponents are explicit.

Definition 1.2 (The source’s proposed value).

Formalization. D5/S3/Arith/Factorization/OeisA343881KageyPowerProducts.A064549 (✓ std3).

Citation. P. Kagey; OEIS Foundation Inc. (2021). OEIS A343881, power-product minima and A064549. URL: https://oeis.org/A343881.

Commentary.

a(n) = n * Product_{primes p|n} p. (OEIS A064549, NAME, revision 71; unpaginated.) The source index n is written k. Nat.primeFactors contains each prime divisor once.

Definition 1.3 (The eventual-equality conjecture).

Formalization. D5/S3/Arith/Factorization/OeisA343881KageyPowerProducts.claim (✓ std3).

Citation. P. Kagey; OEIS Foundation Inc. (2021). OEIS A343881, power-product minima and A064549. URL: https://oeis.org/A343881.

Commentary.

Table read by antidiagonals upward: T(n,k) is the least integer m > k such that k^x * m^y = c^n for some positive integers c, x, and y where x < n and y < n; n >= 2, k >= 1. (OEIS A343881, NAME, revision 16; unpaginated.) Conjecture: T(p,k) = A064549(k) for fixed k > 1 and sufficiently large p. (OEIS A343881, COMMENT, revision 16; unpaginated.) For every natural k greater than one there is a threshold N such that the equality holds for every prime p at least N. Nat.find selects the least larger factor; the value is zero when no such factor exists.

Theorem 1.4 (Refutation at k = 12).

Proof. Machine-checked in Lean as D5/S3/Arith/Factorization/OeisA343881KageyPowerProducts.result (✓ std3). ∎

Resolves. Problems/kagey343881-powerproducts (refuted) by D5/S3/Arith/Factorization/OeisA343881KageyPowerProducts.result.

Source. Repository-derived.

Acknowledgement. P. Kagey; OEIS Foundation Inc. (2021). OEIS A343881, power-product minima and A064549. URL: https://oeis.org/A343881.

Commentary.

At k=12 the proposed value is 72. If 12^x times 72^y were an nth power, its prime coordinates would give n dividing both 2x+3y and x+2y. Twice the latter minus the former is y, so n divides y, contradicting 0<y<n. This obstruction holds for every natural n.

For a prime p a larger factor exists: 27 works at p=2, while 144 works at p at least 3 with exponents p-2 and 1. Thus the least factor itself satisfies Works and cannot be 72. There are primes beyond every threshold, which disproves the eventual equality. The universal equality T(p,12)=144 for primes p at least 7 and a corrected formula for arbitrary k remain unproved here. The separate assertion that prime rows contain distinct integers also remains unresolved.

References