Minimal idempotency, partial idempotency, search heuristics and constructive algorithms for idempotent integers
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 7-21.

Previous work established the set of square-free integers n with at least one factorization n=p ¯q ¯ for which p ¯ and q ¯ produce valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n=p ¯q ¯)(p ¯-1)(q ¯-1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because aZ n ,a k(p ¯-1)(q ¯-1)+1 n a for any positive integer k. This set includes the semiprimes and the Carmichael numbers, but is not limited to them. Numbers in this last category have not been previously analyzed in the literature.

We discuss the structure of idempotent integers here, and present heuristics to assist in finding them. We introduce the notions of partial idempotency and minimal idempotency, give appropriate definitions for them, and present preliminary results.

Un travail antérieur décrit l’ensemble des entiers n sans facteur carré admettant au moins une factorisation n=p ¯q ¯ pour laquelle p ¯ et q ¯ produisent des clés RSA valides, qu’ils soient premiers ou non. Ces entiers sont exactement ceux jouissant de la propriété que λ(n=p ¯q ¯)(p ¯-1)(q ¯-1), où λ est la fonction de Carmichael. Nous appelons ces entiers idempotents, parce que aZ n ,a k(p ¯-1)(q ¯-1)+1 n a pour tout entier positif k. Cet ensemble inclut les nombres semi-premiers et les nombres de Carmichael, mais n’est pas seulement composé de ceux-ci. Les nombres de cette dernière catégorie n’ont pas encore été analysés dans la littérature.

Dans cet article nous discutons la structure des entiers idempotents et présentons des heuristiques pour aider à les trouver. Nous introduisons les notions de partiellement idempotent et d’idempotence minimale, en donnons des définitions appropriées et présentons des résultats préliminaires.

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Published online:
DOI: 10.5802/pmb.53

Barry S. Fagin 1

1 Dept of Computer Science, 2354 Fairchild Drive, US Air Force Academy, Colorado Springs CO 80840
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Barry S. Fagin. Minimal idempotency, partial idempotency, search heuristics and constructive algorithms for idempotent integers. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 7-21. doi : 10.5802/pmb.53. https://pmb.centre-mersenne.org/articles/10.5802/pmb.53/

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