Search results
Results from the WOW.Com Content Network
No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime. With the exception of F 0 and F 1 , the last decimal digit of a Fermat number is 7. The sum of the reciprocals of all the Fermat numbers (sequence A051158 in the OEIS ) is irrational .
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
This page was last edited on 6 July 2024, at 11:25 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply ...
Because of the sparsity of the Fermat numbers, the Pépin test has only been run eight times (on Fermat numbers whose primality statuses were not already known). [ 1 ] [ 2 ] [ 3 ] Mayer, Papadopoulos and Crandall speculate that in fact, because of the size of the still undetermined Fermat numbers, it will take considerable advances in ...
It is currently the second largest known Fermat prime. [1] Analogously, 257 is the third Sierpinski prime of the first kind, of the form + + =. [2] It is also a balanced prime, [3] an irregular prime, [4] a prime that is one more than a square, [5] and a Jacobsthal–Lucas number. [6]
In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In the notation of modular arithmetic , this is expressed as a p ≡ a ( mod p ) . {\displaystyle a^{p}\equiv a{\pmod {p}}.}
When p is a prime, p 2 is a Fermat pseudoprime to base b if and only if p is a Wieferich prime to base b. For example, 1093 2 = 1194649 is a Fermat pseudoprime to base 2, and 11 2 = 121 is a Fermat pseudoprime to base 3. The number of the values of b for n are (For n prime, the number of the values of b must be n − 1, since all b satisfy the ...
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.. A Gaussian integer is a complex number + such that a and b are integers. The norm (+) = + of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer.