Pythagorean triple from Euclid's formula

Statement

Euclid’s formula produces Pythagorean triples $(a,b,c)$ with $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$ (integers $m>n>0$). Using $m=4$ and $n=1$, compute the triple and check that $a^2+b^2=c^2$.

Solution

Substituting $m=4$, $n=1$: $a=4^2-1^2=15,\qquad b=2\cdot 4\cdot 1=8,\qquad c=4^2+1^2=17.$ Check: $15^2+8^2=225+64=289=17^2.$ The triple is $(a,b,c)=(\boxed{15},\ \boxed{8},\ \boxed{17})$.