A primality test for Kpⁿ⁺¹ numbers and a generalization of Safe primes and Sophie Germain primes
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 a modular exponentiation (with a proper base a) similar to that of Fermat's test without the computation of any GCD's. We also provide two tests...
| Published in: | Notes on Number Theory and Discrete Mathematics |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
"Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences
2023-02-01
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| Subjects: | |
| Online Access: | https://nntdm.net/papers/nntdm-29/NNTDM-29-1-062-077.pdf |
| Summary: | 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 a modular exponentiation (with a proper base a) 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 primes. As corollaries, we provide three families of integers N whose primality can be certified only by proving that aᴺ⁻¹ ≣ 1 (mod N) (Fermat's test). One of these families is identical to Safe primes (since N-1 for these integers has large prime factor the same as Safe primes). Therefore, we considered them as a generalization of Safe primes and defined them as a-Safe primes. We address some questions regarding the distribution of those numbers and provide a conjecture about the distribution of their generative numbers a-Sophie Germain primes which seems to be true even if we are dealing with 100, 1000, or 10000 digits primes. |
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| ISSN: | 1310-5132 2367-8275 |