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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.
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 ]
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.
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 ...
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 ...
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 ...
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 ...