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A Pythagorean prime is a prime number of the form 4''n'' + 1. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares. Equivalently, by the Pythagorean theorem, they are the odd prime numbers ''p'' for which is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers ''p'' for which ''p'' itself is the hypotenuse of a Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; is the hypotenuse of a right triangle with legs 1 and 2, and 5 itself is the hypotenuse of a right triangle with legs 3 and 4. ==Values and density== The first few Pythagorean primes are :5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, ... . By Dirichlet's theorem on arithmetic progressions, this sequence is infinite. More strongly, for each ''n'', the numbers of Pythagorean and non-Pythagorean primes up to ''n'' are approximately equal. However, the number of Pythagorean primes up to ''n'' is frequently somewhat smaller than the number of non-Pythagorean primes; this phenomenon is known as Chebyshev's bias.〔.〕 For example, the only values of ''n'' up to 600000 for which there are more Pythagorean than non-Pythagorean odd primes are 26861 and 26862.〔 〕 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「Pythagorean prime」の詳細全文を読む スポンサード リンク
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