HINT: $a^2+b^2=c^2$ cannot hold for $a,b$ odd and $c$ even. (using congruences)
127891
If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even.
17047
Prove that if n is even then $n^2$ is even and if n is odd then $n^2$ is odd?
137015
Solutions for $a^2+b^2+c^2=d^2$
67666
show that $a^2+b^2+c^2+2abc+3\ge (1+a)(1+b)(1+c)$
20834
Integer solutions to $c=a^2-b^2$
115460
Proving $a\mid b^2,\,b^2\mid a^3,\,a^3\mid b^4,\ldots\implies a=b$ - why is my approach incorrect
131941
Solve quadratic congruences.
152039
Prove that $a^{2^n}=1 \mod 2^{n+2}$
150085
Solutions to the diophantine equation: $2a^2 + 2b^2- c^2- d^2 = 0$
37808
Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
50614
If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$
79219
Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
100913
It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be a perfect square
65297
Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$.
125164
How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
108035
solving diophantine equations using congruences
142133
Prove that if $a^2 + b^2 = c^2$, then 5 divides abc. a,b,c are positive integers.