Diophantine Equation #2

Number Theory Level 5

Consider the following diophantine equation:

$\displaystyle x^2 + xy + y^2 = n$

For a particular positive integer $n$ , the number of solutions $(x, y)$ such that $x$ and $y$ are integers is given by the function $S(n)$ .

The function $S(n)$ is not one-to-one. For example, each number $n$ in the set $\{1, 4, 9, 16, 25, 36\}$ corresponds with $S(n) = 6$ .

In increasing order, starting from $n = 1$ , the first $n$ such that $S(n) = 36$ is $637$ .

What is the 500th $n$ such that $S(n) = 36$ ?

The answer is 88387.

1 solution

Yuriy Kazakov
Jul 25, 2020

I find A004016 and Set solution number and see that solution exist for $n=m^2 3^r\prod {p_k}^{n_{k}}$ where $p_k =3q+1$ - prime number and $m$ has not prime divisor form $3q+1$ . And $S(n)=36$ for $n= m^2 3^r {p_k}^{2}{p_s}$ - form $1$ or $n= m^2 3^r {p_k}^{5}$ - form $2$ .

And I use Python and find $497$ numbers of form $1$ and $3$ numbers of form $2$ . Answer $\boxed {88387}$ .

Diophantine Equation #2

The answer is 88387.

1 solution

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