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In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property. The sequence of integer congruent numbers starts with : 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, … For example, 5 is a congruent number because it is the area of a 20/3, 3/2, 41/6 triangle. Similarly, 6 is a congruent number because it is the area of a 3,4,5 triangle. 3 is not a congruent number. If ''q'' is a congruent number then ''s''2''q'' is also a congruent number for any natural number ''s'' (just by multiplying each side of the triangle by ''s''), and vice versa. This leads to the observation that whether a nonzero rational number ''q'' is a congruent number depends only on its residue in the group :. Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers, when speaking about congruent numbers. ==Congruent number problem== The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2012) been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven. Fermat's right triangle theorem, named after Pierre de Fermat, states that no square number can be a congruent number. However, in the form that every congruum (the difference between consecutive elements in an arithmetic progression of three squares) is non-square, it was already known (without proof) to Fibonacci.〔.〕 Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number.〔.〕 However, determining whether a number is a congruum is much easier than determining whether it is congruent, because there is a parameterized formula for congrua for which only finitely many parameter values need to be tested.〔.〕 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「congruent number」の詳細全文を読む スポンサード リンク
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