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In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form : where ''n'' is a nonnegative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … . If 2''k'' + 1 is prime, and ''k'' > 0, it can be shown that ''k'' must be a power of two. (If ''k'' = ''ab'' where 1 ≤ ''a'', ''b'' ≤ ''k'' and ''b'' is odd, then 2''k'' + 1 = (2''a'')''b'' + 1 ≡ (−1)''b'' + 1 = 0 (mod 2''a'' + 1). See below for a complete proof.) In other words, every prime of the form 2''k'' + 1 (other than 2 = 20 + 1) is a Fermat number, and such primes are called Fermat primes. As of 2015, the only known Fermat primes are ''F''0, ''F''1, ''F''2, ''F''3, and ''F''4 . ==Basic properties== The Fermat numbers satisfy the following recurrence relations: : for ''n'' ≥ 1, : : : for ''n'' ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ ''i'' < ''j'' and ''F''''i'' and ''F''''j'' have a common factor ''a'' > 1. Then ''a'' divides both : and ''F''''j''; hence ''a'' divides their difference, 2. Since ''a'' > 1, this forces ''a'' = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each ''F''''n'', choose a prime factor ''p''''n''; then the sequence is an infinite sequence of distinct primes. Further properties: *No Fermat prime can be expressed as the difference of two ''p''th powers, where ''p'' is an odd prime. *With the exception of F0 and F1, the last digit of a Fermat number is 7. * The sum of the reciprocals of all the Fermat numbers is irrational. (Solomon W. Golomb, 1963) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fermat number」の詳細全文を読む スポンサード リンク
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