query: Prove that number $4p^2+1$ can show as sum of squares of three different numbers
positive_passages.docid: 146512
positive_passages.text: Prove that for every prime number $p>3$, $4p^2+1$ can be written as the sum of three square numbers
negative_passages.docid: 32350
negative_passages.text: Compilation of proofs for the summation of natural squares and cubes
negative_passages.docid: 86285
negative_passages.text: Given positive integers $ x , y$ and that $3x^2 + x = 4y^2 +y$ Prove that $x-y$ is a perfect square
negative_passages.docid: 144335
negative_passages.text: Sum of three numbers and the sum of their squares: $x^2+y^2+z^2 \geq \frac{1}{3}$ for $x+y+z=1$
negative_passages.docid: 20443
negative_passages.text: Calculate sum of squares of first $n$ odd numbers
negative_passages.docid: 35342
negative_passages.text: Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
negative_passages.docid: 78917
negative_passages.text: Show that for any prime numbers $p,q,r$, one has $p^2+q^2 \ne r^2$.
negative_passages.docid: 146333
negative_passages.text: Gaussian proof for the sum of squares?
negative_passages.docid: 88036
negative_passages.text: Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
negative_passages.docid: 113359
negative_passages.text: Proving that $x^2 + 4$ is not divisible by $3$
negative_passages.docid: 99606
negative_passages.text: Prove or disprove there exists a set of four numbers out of the $48$ positive integers of which product is a perfect square
negative_passages.docid: 4333
negative_passages.text: Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$
negative_passages.docid: 50615
negative_passages.text: Expressing a Non Negative Integer as Sums of Two Squares
negative_passages.docid: 75318
negative_passages.text: How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?
negative_passages.docid: 11506
negative_passages.text: Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer
negative_passages.docid: 90519
negative_passages.text: Combinatorial proof of $\sum\limits_{k=0}^n {n \choose k}3^k=4^n$
negative_passages.docid: 40565
negative_passages.text: How can I prove that one of $n$, $n+2$, and $n+4$ must be divisible by three, for any $n\in\mathbb{N}$