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In mathematics, a Fermat number, ... That 641 is a factor of F 5 can be deduced from the equalities 641 = 2 7 × 5 + 1 and 641 = 2 4 + 5 4.
This pattern breaks down after this, as the next Fermat number is composite (4294967297 = 641 × 6700417), so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, and it is therefore unknown how many odd-sided constructible regular polygons exist.
Fermat conjectured that all numbers of the form + (known as Fermat numbers) were prime. However, this conjecture was disproved by Euler , who found that 2 2 5 + 1 = 4 , 294 , 967 , 297 = 641 × 6 , 700 , 417. {\displaystyle 2^{2^{5}}+1=4,294,967,297=641\times 6,700,417.} [ 23 ]
The probability of a composite number n passing the Fermat test approaches zero for . Specifically, Kim and Pomerance showed the following: The probability that a random odd number n ≤ x is a Fermat pseudoprime to a random base 1 < b < n − 1 {\displaystyle 1<b<n-1} is less than 2.77·10 −8 for x= 10 100 , and is at most (log x) −197 <10 ...
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...
A prime number (or prime) ... 641: 643: 647: 653: 659: ... As of June 2024 these are the only known Fermat primes, and conjecturally the only Fermat primes. The ...
In number theory, a Wagstaff prime is a ... 31, 53, 67, 293, 641, 2137, 3011, 268207, ... (sequence A001562 ... Wagstaff and Fermat numbers based on cycles of the ...
In number theory, a regular prime is a ... Vandiver proved the first case of Fermat's Last Theorem for E-regular primes), ... 641 59 44 157 62, 110 269 389 200 509 ...