The congruent number problem is the problem of determining whether a given rational number is or is not a congruent number.
Since $n$ is congruent if and only if $nt^2$ is, for any nonzero rational number $t$, it is sufficient to consider the case where $n$ is a square-free integer; some sources state the congruent number problem only for positive integers for this reason.
A positive integer $n$ is congruent if and only if there exists a rational number $a$, such that $a$, $a-n$ and $a+n$ are all squares of rational numbers. For, given rational $X,Y,Z$ satisfying $X^2+Y^2=Z^2$ and $XY=2n$, the rational number $a=(Z/2)^2$ satisfies $a\pm n=((X\pm Y)/2)^2$, and conversely, given $a=A^2$ with $a+n=B^2$ and $a-n=C^2$, set $(X,Y,Z)=(2A,B+C,B-C)$.
For example, with the $3,4,5$ right triangle and $n = 6$, let $a = 25/4=(5/2)^2$; then $a-n=25/4 - 6 = (1/2)^2$ and $a+n=25/4 + 6 = (7/2)^2$.
It is known that a positive integer number $n$ is congruent if and only if there exists a rational solution $(x,y)$ with $y\not=0$ to the elliptic curve equation: \[ E_n:\quad y^2 = x^3 - n^2 x, \] or, in other words, a rational point $P=(x,y)$ on the congruent number curve $E_n(\Q)$ with $y\not=0$.
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- Last edited by John Cremona on 2021-07-14 09:14:46
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