Show that $q\equiv_8 1$ when $q$ is an odd square number
7414
Prove that odd perfect square is congruent to $1$ modulo $8$
12729
Proving $9^n - 8n - 1$ is divisible by $8^2$ for $n\ge 0$?
134963
Proving a non-prime (composite) number $> 1$ has a divisor $d \leq\sqrt q$
118629
$p,q$ odd primes, $q\mid a^p-1\Rightarrow q\mid a-1$ or $\,2p\mid q-1$
95111
Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
114796
Direct proof that the product of odd integers is odd
74512
How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
77816
Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
92166
Number of odd binomial coefficients is a power of $2$
144851
For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
513
If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.
72054
Show that $\deg f$ is even when $n$ is odd.
41641
$(n-7)(n+7)=$ some perfect square
12862
Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd
148774
show elements of order 15 in $S_8$ satisfy a relation
28957
Verifying that $2^{44}-1$ is divisible by $89$
48787
A number is a perfect square if and only if it has odd number of positive divisors