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A congruent number $n$ is a positive rational number which is the area $\frac{1}{2}XY$ of a right triangle with rational legs $X$, $Y$ and rational hypotenuse $Z$, so $X,Y,Z$ are positive rational numbers satisfying the equations \[ X^2 + Y^2 = Z^2 \tag{1} \] and \[ \frac{1}{2} XY = n. \tag{2} \]

For example, $6$ is a congruent number because it is the area of the $3,4,5$ right triangle. So is $5$, with $(X,Y,Z)=(3/2,20/3,41/6)$, while none of the integers $1$, $2$, $3$ or $4$ is a congruent number. The integers less than $100$ which are congruent are \[ 5,6,7,13,14,15,20,21,22,23,24,28,29,30,31,34,37,38,39,41,45,46,47, 52,53,54,55,56,60,61,62,63,65,69,70,71,77,78,79,80,84,85,86,87,88,92,93,94,95,96\]

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  • Review status: beta
  • Last edited by John Cremona on 2021-07-14 09:03:30
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