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19 votes
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univariate integer version of Hilbert's 17th problem

Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
Fedor Petrov's user avatar
14 votes
0 answers
481 views

If $ab^2$ is a sum of three squares, then so is $a$. How to see it quickly?

Here $a, b$ are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way?
Fedor Petrov's user avatar
3 votes
0 answers
338 views

The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$: $$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$ This is A001923 in the OEIS. I don't have much experience with ...
Zubin Mukerjee's user avatar
0 votes
0 answers
196 views

Sum of squares squared in an arithmetic progression

Let $r(n)$ be the number of ways to write $n$ as a sum of two squares and $(a,q)=1$. What is known about $$ \sum_{n \le x,n \equiv a (\text{mod} \, q)} r(n)^2 \quad? $$ I am looking for uniform ...
toshi's user avatar
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