Number Theory

    

On Some Conjectures Concerning Perfect Powers

Authors: Marco Ripa

The starting point of our paper is Kashihara’s open problem #30, concerning the sequence A001292 of the OEIS, asking how many terms are perfect squares. We confirm his last conjecture up to the 100128-th term and provide a general theorem which rules out 4/9 of the candidates. Moreover, we formulate a new, provocative, conjecture involving the OEIS sequence A352991 (which includes all the terms of A001292, except the first one). Our risky conjecture states that all the perfect powers belonging to the sequence A352991 are perfect squares and they cannot be written as higher order perfect powers. This challenging conjecture has been checked for any integer smaller than 10111121314151617181920212223456789 and no counterexample has been found so far.

Comments: 7 Pages.

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Submission history

[v1] 2022-04-23 19:38:23
[v2] 2022-04-29 18:10:32
[v3] 2022-05-15 19:46:14

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