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Kotěšovec’s second monomial-sum conjecture

Abstract

Every biquadratefree powerful number, and each of 243, 972, 1944 and 2401, has nonzero complete monomial sums at every degree at least two.

Definition 1.1 (The second conjecture).

Formalization. D5/S3/Arith/CharacterSums/KotesovecConjectureTwo.claimTwo (✓ std3).

Citation. Václav Kotěšovec (2024). OEIS A368959: nonzero complete monomial exponential sums. URL: https://oeis.org/A368959.

Commentary.

OEIS A368959, COMMENTS: “Conjecture 2: This sequence contains all the terms from A338325, plus 243, 972, 1944, 2401, …” The ellipsis is not an exhaustive enumeration: the proposition asserts the inclusion and the four named terms. The predicates InA338325 and InA368959 are those defined in KotesovecConjectureOne.

Theorem 1.2 (The family and four further moduli).

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

Resolves. Problems/kotesovec-2024-a368959-family (proved) by D5/S3/Arith/CharacterSums/KotesovecConjectureTwo.result.

Source. Repository-derived.

Acknowledgement. Václav Kotěšovec (2024). OEIS A368959: nonzero complete monomial exponential sums. URL: https://oeis.org/A368959.

Commentary.

Nonvanishing for injective additive characters holds for prime squares, prime cubes, 243 and 2401. Chinese remaindering preserves it for coprime products. The products 972=4243 and 1944=8243 yield the remaining two moduli, and the standard additive character gives the exponential sums of A368959.

References