Prove that for any integers $a, b, c, d$ number $(a^2 + b^2)(c^2 + d^2)$ is a sum of two squares of integer.
112644
The product of two numbers that can be written as the sum of two squares
80148
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
93657
Proving “The sum of $n$ consecutive cubes is equal to the square of the sum of the first $n$ numbers.”
24551
How to prove that the sum and product of two algebraic numbers is algebraic?
59479
For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
128199
Proving gcd(a, a+2) is 1 or 2 for every integer a?
127566
$2AB$ is a perfect square and $A+B$ is not a perfect square
11620
Is it true that any matrix in $M_2(\mathbb R)$ is the sum of two squares?
132098
Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
131625
Expressing an integer as the sum of three squares
6621
Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$
99692
Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$
100571
Prove $(1-a)(1-b)(1-c)(1-d)\geq abcd$ if $a^2+b^2+c^2+d^2=1$
141102
If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
14480
Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$
67332
Sum of two squares modulo p
23835
Discrete mathematics - Find all integer solutions of the equation $a^2+ b^2 + c^2=a^2 b^2$.