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Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.

Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,x_n)=K$ subject to:

  • the primes divisors of $x_i$ and $K$ all lie in $S$ and

  • each $x_i$ divides $K$?

I would also be interested in the more general version where we allow positive integer coefficients.

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    $\begingroup$ S-integers usually mean rational numbers with denominators divisible only by primes in S. That is not what you are talking about in the body of your question. Also, what is the question? $\endgroup$
    – Felipe Voloch
    Commented May 16, 2014 at 15:37
  • $\begingroup$ If someone has studied the problem for S-integers then this will help as I can clear denominators. I have added a ? and a tag to make it clearer. $\endgroup$
    – Eric Rowell
    Commented May 16, 2014 at 17:37
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    $\begingroup$ S-integers in the usual sense are allowed any integer numerators. What you are talking about is, I think, S-units. The S-unit equation (sum of S-units = 1) has been studied a lot with the main result being finiteness of solutions (modulo obvious conditions). You still haven't stated a question. $\endgroup$
    – Felipe Voloch
    Commented May 16, 2014 at 17:41
  • $\begingroup$ I am aware of the result of Evertse and others showing that there are finitely many solutions, even in the case of number fields. I want to find all solutions if possible. $\endgroup$
    – Eric Rowell
    Commented May 16, 2014 at 20:45

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