If $n$ is an odd integer, then there exist integers $a$ and $b$ such that $n=a^2-b^2$.
94570
Proof that every odd integer is a difference of two squares
95588
Prove by Contradiction: if $4\mid a^2 + b^2$, then $a$ and $b$ are not both odd
92166
Number of odd binomial coefficients is a power of $2$
52089
Group of $2n$ elements, $n$ odd, is not simple
92057
Prove that for every odd integer n, there is a group...
17047
Prove that if n is even then $n^2$ is even and if n is odd then $n^2$ is odd?
63661
integers $n$ such that $(n-1)!+1$ is a power of $n$
155784
If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers
122612
An integer is a $n$'th power if that holds true for all moduli.
56574
For any integer $a,b$ let $N_{a,b}$ denote the number of positive integer $x<1000$ satisfying $x= a( mod\;27)$ and $x=b(mod\;37)$. Then
113168
If $a > b$, is $a^2 > b^2$?
95162
For a odd prime $p, \exists a,b$ such that $a^2 + ab + b^2 \equiv 0 \pmod{p} \iff \exists x,y $ such taht $x^2 + xy + y^2 = p$?
102154
If $a$ and $b$ are consecutive integers, prove that $a^2 + b^2 + a^2b^2$ is a perfect square.
142724
Prove that for all integers $x$ and $y$, $x - y$ is odd if and only if $x + y$ is odd.
44182
Proving that an integer is even if and only if it is not odd
31895
Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$
48404
If $a$ and $b$ are integers such that $9$ divides $a^2 + ab + b^2$ then $3$ divides both $a$ and $b$.