Can You Find Integer Solutions? | Diophantine Equations | Math Olympiad Training

Find all right triangles with integer side lengths such that their areas and perimeter are equal.

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We can write a pythagorean triple as r^2-k^2, 2rk, r^2+k^2



r^3k-k^3r=2r^2+2rk
divide by r
r^2k-k^3=2r+2k
kr^2-2r-2k-k^3=0
(2+sqrt(4+4k(2k+k^2)))/2k
(1+sqrt(1+k(2k+k^2)))/k
1+2k^2+k^3 is a perfect square
It is 1 mod k
in sqrt, we take the positive root, but it must divide k, so 1 is congruent to -1 mod k
True for k=1, 2
for k=1, r=3, a=8, b=6, c=10, 8*6/2=24=8+6+10

for k=2, r=3, a=5, b=12, c=13, 5*12/2=5+12+13