2022/07/25 by A. Ramzy, Ramzy, A.
Mathematics · #11N05 #11N80 #11Y11 #Analytic Number Theory Research #Benford’s Law and Fraud Detection #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2207.12407
openalex publication_date 2022/07/25 · openalex created_date 2023/02/24 · openalex updated_date 2026/07/28
In this paper, we provide a generalization of Proth's theorem for integers of the form Kpn+1. In particular, a primality test that requires only one modular exponentiation similar to that of Fermat's test without the computation of any GCD's. We also provide two tests to increase the chances of proving the primality of Kpn+1 numbers (if they are primes indeed). As a corollaries of these tests we provide three families of integers N whose primality can be certified only by proving that aN-1 ≡ 1 \pmod N (Fermat's test). We also generalize Safe Primes and define those generalized numbers as a-SafePrimes for being similar to SafePrimes (since N-1 for these numbers has large prime factor the same as SafePrimes), we address some questions regarding the distribution of those numbers and provide a conjecture about the distribution of their related numbers a-SophieGermainPrimes which seems to be true even if we are dealing with 100, 1000, or 10000 digits primes.