Computers as Novel Mathematical Reality. VI. Fermat numbers and their relatives
نویسندگان
چکیده
In this part, which constitutes a pendent to the part dedicated Mersenne numbers, I continue discuss fantastic contributions towards solution o classical problems of number theory achieved over last decades with use computers. Specifically, address primality testing, factorisations and search prime divisors numbers certain special form, primarily Fermat their friends relations, such as generalised Proth like. Furthermore, we role primes Pierpoint in cyclotomy.
منابع مشابه
Mersenne and Fermat Numbers
The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127 —1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in ...
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Fn+1 = (Fn − 1) + 1, n ≥ 0 and are pairwise coprime. It is conjectured that Fn are always square-free and that, beyond F4, they are never prime. The latter would imply that there are exactly 31 regular polygons with an odd number Gm of sides that can be constructed by straightedge and compass [2]. The values G1, G2, . . ., G31 encompass all divisors of 232−1 except unity [3]. Let G0 = 1. If we ...
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A search for prime factors of the generalized Fermat numbers Fn(a, b) = a2 n + b2 n has been carried out for all pairs (a, b) with a, b ≤ 12 and GCD(a, b) = 1. The search limit k on the factors, which all have the form p = k · 2m + 1, was k = 109 for m ≤ 100 and k = 3 · 106 for 101 ≤ m ≤ 1000. Many larger primes of this form have also been tried as factors of Fn(a, b). Several thousand new fact...
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ژورنال
عنوان ژورنال: ???????????? ??????????? ? ???????????
سال: 2022
ISSN: ['2071-2359', '2071-2340']
DOI: https://doi.org/10.32603/2071-2340-2022-4-5-67