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A simple note on Fermat numbers, on Mersenne numbers and on numbers of the form bn, where bn = 2 n + 1 or bn = 2 n+1 +...
ISSN: 2195-1381
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A simple note on Fermat numbers, on Mersenne numbers and on numbers of the form bn, where bn = 2 n + 1 or bn = 2 n+1 +...
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Abstract
A Fermat number is a number of the form Fn = 2n+ 1, where n is an integer ≥ 0. A Fermat composite (see [1] or [2] or [4] ) is a non prime Fermat number and a Fermat prime is a prime Fermat number. Fermat composites and Fermat primes are characterized via divisibility in [4] and in [5]. It is known (see [4] ) that for every j ∈ {0, 1, 2, 3, 4}, Fj is a Fermat prime and it is also known (see [2] or [3]) that F5 and F6 are Fermat composites. A Mersenne number is a number of the form Mm = 2m − 1 [where m is an integer ≥ 0]. A Mersenne composite (see [3] or [5] ) is a non prime Mersenne number. Mersenne composites and Mersenne primes are characterized via divisibility in [5] [ A Mersenne prime is a prime Mersenne number (see [3] or [5]); it is immediate to see that if Mn = 2 n − 1 is a Mersenne prime, then n is prime and therefore there are infinitely many Mersenne composite numbers] . It is known (see [3]) that M11 and M67 are Mersenne co...
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References
[1] Dickson.Theory of Numbers (History of Numbers. Divisibity and pri-mality) Vol1. Chelsea Publishing Company. New York , N.Y (1952).Preface.IIIto Preface.XII.
[2] G.H Hardy, E.M Wright.An introduction to the theory of numbers. Fith Edition. Clarendon Press. Oxford.
Paul Hoffman. Erd¨os, l’homme qui n’aimait que les nombres. Editions Belin, (2000). 30 − 49.
[4] Ikorong Annouk.Placed Near The Fermat Primes And The FermatComposite Numbers. International Journal Of Research In Mathematic AndApply Mathematical Sciences; Vol3; 2012, 72−82.
[5] Ikorong Annouk.Then We Characterize Primes and Composite NumbersVia Divisibility. International Journal of Advanced In Pure MathematicalSciences; Volume 2, no.1; 2014.
