Prove that there aren't any positive integers $a,b>2$ such that $a^2 \vert b^3+1$ and $b^2 \vert a^3+1$
58133
${a^2}{\mid}({b^3} + 1)\;$and$\;{b^2}{\mid}({a^3} + 1)$
104920
Can we find positive integers $a$ and $k \geq 2$ with $2^n - 1 = a^k$?
125164
How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
63651
If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax+by=1$.
55582
How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
66261
Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$
134834
$n^2 + (n+1)^2 = m^3$ has no solution in the positive integers
23301
There does not exist any integer $m$ such that $3n^2+3n+7=m^3$
135955
Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis)
138894
Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
67527
Proof by Induction that $3^n ≥ 1+2^n$
97750
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.
145558
Prove that $2^n+3^n $ is never a perfect square
95490
What are the positive integer solutions to $x^2-x+1 = y^3$?
153313
Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
50663
prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
11506
Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer