Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Is there software that helps list small solutions of the Diophantine equation $$ x_0^2=1+x_1^2+x_2^2+\cdots+ x_n^2 $$ where "small" is negotiable, but e.g. we could fix $x_0$ and and ask for the list of all solutions $x_1, \dots, x_n\in\mathbb Z$?

share|cite|improve this question
This is $$(x_0-x_1)(x_0+x_1)=x_0^2-x_1^2=1+x_2^2+\cdots +x_n^2,$$ so sort of reduces to factorization. –  Robin Chapman Jun 10 '10 at 20:17
@Will, I wish to distinguish $x_j$ and $-x_j$. Please do not bother to write anything in C++; I just want to know whether there is anything already available as part of Mathematica ot other similar package. @Robin, thanks, reducing factorization is helpful idea. –  Igor Belegradek Jun 10 '10 at 21:01
For $n=2$ you can try for various values of $x_0$. –  Eric Rowell Jun 10 '10 at 21:12
SquaresR[d, n] gives the number of ways r_d (n) to represent the integer n as a sum of d squares. PowersRepresentations[n, k, p] gives the distinct representations of the integer n as a sum of k non-negative p\^th integer powers. EllipticTheta[a,u,q] gives the theta function Subscript[[CurlyTheta], a](u,q) (a=1,[Ellipsis],4). –  Will Jagy Jun 10 '10 at 21:19
note the above are Mathematica commands –  Will Jagy Jun 10 '10 at 21:52

2 Answers 2

Have you had a look on this tutorial: ? To take different values n < n_max into account, a simple loop could work.

share|cite|improve this answer

This function in Mathematica find them all:

f[n_] := Reduce[ Total[Table[x[i]^2, {i, 1, n}]] + 1 == x[0]^2 && 
             (Table[x[i], {i, 1, 3}] /. List -> LessEqual), 
              Table[x[i], {i, 0, n}], Integers]   

Invoke with:

f[1], f[3] ... etc
share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.