Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers

Previous work established the set of square-free integers n with at least one factorization n=p¯q¯ for which p¯ and q¯ are valid RSA keys, whether they prime or composite. These exactly those property λ(n)∣(p¯−1)(q¯−1), where λ is Carmichael totient function. We refer to these as idempotent, because ∀a∈Zn,ak(p¯−1)(q¯−1)+1≡na any positive integer k. This was initially known contain only semiprimes, later expanded include some numbers. Recent by author gave explicit formulation set, showing that includes numbers neither semiprimes nor Numbers in this last category had not been previously analyzed literature. While have useful cryptographic properties, idempotent deserving study their own right lie border hard problems number theory computer science. Some integers, maximally all factorizations idempotent. discuss structure here, heuristics assist finding them, algorithms from graph can be used construct examples arbitrary size.