# Questions tagged [sums-of-squares]

For questions concerning various representation of integers as sums of squares, which are studied in number theory.

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### Largest $k$ such that $(p-k) = \lceil \sqrt{(p-2k) p} \rceil$

Assume $p \in \mathbb P.$ Assume $0<p-2k<p$ and the next square larger than $p(p-2k)$ is $(p-k)^2$. It is trivial to show that $p(p-2k)+k^2$ is a square. Simply $p(p-2k)+k^2 = (p-k)^2.$ ...
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### A four-digit square is of the form $aabb$. What's $a^2+b^2$? [duplicate]

The 5772 Ulpaniada included the following question: Consider a four digit square number (a number which is the square of a whole number).Its digit notation is $aabb$ (the thousands digit is $a$, ...
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### Algorithm to find solution to $ax^2 + by^2 = 1$ in a finite field

Let $\mathbb{F}$ be a finite field, and let $a,b \in \mathbb{F}$ be given, subject to $a\ne 0, b \ne 0$. Consider the equation $$ax^2 + by^2 = 1.$$ It is guaranteed that there exists a solution to ...
### For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ nonzero squares for all $1 \leq i \leq n$?
I need to prove this result for something I am working on: For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ nonzero squares for all \$1 \leq ...