Pseudoprimes

Euler pseudoprime

In arithmetic, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and (where mod refers to the modulo operation). The motivation for this definition is the fact that all prime numbers p satisfy the above equation which can be deduced from Fermat's little theorem. Fermat's theorem asserts that if p is prime, and coprime to a, then ap−1 ≡ 1 (mod p). Suppose that p>2 is prime, then p can be expressed as 2q + 1 where q is an integer. Thus, a(2q+1) − 1 ≡ 1 (mod p), which means that a2q − 1 ≡ 0 (mod p). This can be factored as (aq − 1)(aq + 1) ≡ 0 (mod p), which is equivalent to a(p−1)/2 ≡ ±1 (mod p). The equation can be tested rather quickly, which can be used for probabilistic primality testing. These tests are twice as strong as tests based on Fermat's little theorem. Every Euler pseudoprime is also a Fermat pseudoprime. It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves. The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or Carmichael numbers, and the smallest absolute Euler pseudoprime is 1729 = 7×13×19. (Wikipedia).

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Related pages

Probable prime | Strong pseudoprime | Fermat's little theorem | Number | Euler–Jacobi pseudoprime | Lua (programming language) | Jacobi symbol | Greatest common divisor | Odd number | Pseudoprime | Modular exponentiation | Carmichael number | Composite number | Integer | Fermat pseudoprime | Subset | Prime number | Arithmetic | Modular arithmetic