Fermat’s Little Theorem
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Extension and Generalization of Fermat’s Little Theorem to the Gaussian Integers
It can . . . come as a bit of a shock to meet your first non-obvious theorem, which will typically be Fermat’s Little Theorem. — Dominic Yeo [10] The non-obviousness of Fermat’s Little Theorem is the most interesting part of any introductory number theory course. We are therefore motivated to determine if Fermat’s Little Theorem can be extended to the Gaussian integers, as many other useful pro...
متن کاملGeneralizations of Fermat's Little Theorem via Group Theory
Let p be a prime number and a be an integer. Fermat’s little theorem states that a ≡ a (mod p). This result is generally established by an appeal to the theorem of elementary group theory that asserts that x|G| = 1 for every element x of a finite group G. In this note we describe another way that group theory can be used to establish Fermat’s little theorem and related results.
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In this survey, we describe three algorithms for testing primality of numbers that use Fermat’s Little Theorem.
متن کاملFermat’s Test
Fermat’s little theorem says for prime p that ap−1 ≡ 1 mod p for all a 6≡ 0 mod p. A naive extension of this to a composite modulus n ≥ 2 would be: for a 6≡ 0 mod n, an−1 ≡ 1 mod n. Let’s call this “Fermat’s little congruence.” It may or may not be true. When n is prime, it is true for all a 6≡ 0 mod n. But when n is composite it usually has many counterexamples. Example 1.1. When n = 15, the t...
متن کاملIntroductory Number Theory
We will start with introducing congruences and investigating modular arithmetic: the set Z/nZ of “integers modulo n” forms a ring. This ring is a field if and only if n is a prime number. A study of the multiplicative structure leads to Fermat’s Little Theorem (for prime n) and to the Euler phi function and Euler’s generalization of Fermat’s Theorem. Another basic tool is the Chinese Remainder ...
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تاریخ انتشار 2014