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In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers ''D'' such that any integer expressible in only one way as ''x''2 ± ''Dy''2 (where ''x''2 is relatively prime to ''Dy''2) is a prime, prime power, twice one of these, or a power of 2. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes. A positive integer ''n'' is idoneal if and only if it cannot be written as ''ab'' + ''bc'' + ''ac'' for distinct positive integer ''a, b'', and ''c''.〔Eric Rains, Comments on A000926, December 2007.〕 It is sufficient to consider the set ; if all these numbers are of the form , , or ''2''s for some integer s, where is a prime, then is idoneal.〔Roberts, Joe: The Lure of the Integers. The Mathematical Association of America, 1992〕 The 65 idoneal numbers found by Carl Friedrich Gauss and Leonhard Euler and conjectured to be the only such numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 . In 1973, Peter J. Weinberger proved that at most one other idoneal number exists, and that the list above is complete if the generalized Riemann hypothesis holds.〔(Acta Arith., 22 (1973), p. 117-124 )〕 ==See also== *List of unsolved problems in mathematics 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「Idoneal number」の詳細全文を読む スポンサード リンク
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