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  "body": "\n## Problem\n\nA positive integer $n$ is a *congruent number* if there is a right triangle with rational side lengths and area $n$.\nThe problem is to decide, for a given positive integer $n$, whether $n$ is a congruent number.\n\n## Remarks\n\nThe problem reduces to [[square-free|08X0VCR2NW]] $n$, since $n$ is congruent if and only if $nk^2$ is congruent for every positive integer $k$.\nThe number $6$ is congruent, by the triangle with sides $3, 4, 5$, which is a [[Pythagorean triple|0M7ARED5Q7]].\nThe number $5$ is congruent: the sides $3/2$, $20/3$, $41/6$ satisfy $(3/2)^2 + (20/3)^2 = 1681/36 = (41/6)^2$, and the area is $\\frac12 \\cdot \\frac32 \\cdot \\frac{20}{3} = 5$.\nThe number $1$ is not congruent; this is [[Fermat's right triangle theorem|086BE0J3FZ]].\nA positive integer $n$ is congruent if and only if the [[elliptic curve|427N6S6QQW]] $y^2 = x^3 - n^2 x$ has a rational point with $y \\ne 0$.\nEquivalently, this curve has positive rank.\nJerrold Tunnell gave in 1983 an explicit criterion by counting integer solutions of certain ternary quadratic equations.\nThe criterion is necessary for $n$ to be congruent, and its sufficiency follows from the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].\nSo the problem is open in general.\n\n## Explanations\n\n### L1 human\n\nThe question is simple: which whole numbers can be the area of a right triangle whose three sides are fractions?\nFor $6$ the familiar $3$-$4$-$5$ triangle works, and for $5$ a triangle with sides $3/2$, $20/3$, $41/6$ works.\nFor $1$, $2$ and $3$ no such triangle exists.\nThe surprising part is that the question is really about rational points on an elliptic curve.\nA complete answer is tied to one of the major open conjectures of number theory.\n",
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