Re: Does x^2+y^2=z^50 have integer solutions?

Bill Dubuque wrote:

Timothy Murphy <gayleard@xxxxxxxxxx> wrote:
alvarez.torres1@xxxxxxxxx wrote:

Does x^2 + y^2 = z^50 have integer solutions >0?

A positive integer n is expressible in the form n = x^2 + y^2
if and only if each prime q = 3 mod 4 occurs to an even power in n.
(See eg Hardy & Wright, "An introduction to the theory of numbers".)

It follows that your equation has a solution for _every_ z.

That's trivially true: 0^2 + (z^25)^2 = z^50, so there's
no need to infer it from theorem in the prior paragraph.

(Well, you need to exclude some z if you want both x,y > 0.)

Since the problem requires solutions >0, your remark doesn't help.

I wasn't purporting to give a complete answer to the question.
I was just pointing out that it reduces to the question
of which n are expressible in the form x^2 + y^2.
I mentioned Hardy & Wright,
reference to which would answer the question completely.

If you want a complete answer, n is expressible as x^2 + y^2
with x,y > 0 iff each prime q = 3 mod 4 occurs to an even power,
and some prime not = 3 mod 4 occurs in n.

So the problem is soluble with any non-zero z
divisible by some prime p not = 3 mod 4.



.

Relevant Pages

  • Re: Does x^2+y^2=z^50 have integer solutions?
    ... if and only if each prime q = 3 mod 4 occurs to an even power in n. ... (See eg Hardy & Wright, "An introduction to the theory of numbers".) ... reference to which would answer the question completely. ...
    (sci.math)
  • Re: Does x^2+y^2=z^50 have integer solutions?
    ... Timothy Murphy wrote: ... if and only if each prime q = 3 mod 4 occurs to an even power in n. ... (See eg Hardy & Wright, "An introduction to the theory of numbers".) ...
    (sci.math)
  • Re: Transcendental Dimensions
    ... >> There were no 1979 publications by Hardy & Wright. ... If a person can't be a pedant in sci.math, ...
    (sci.math)
  • Re: Does x^2+y^2=z^50 have integer solutions?
    ... I mentioned Hardy & Wright, ... Thus, based on your observations, I think the following iff condition ... Let n be a positive integer. ...
    (sci.math)
  • Re: Farey Fractions and "reducibility"
    ... > three successive terms in F_n, ... for perhaps one has to reduce the RHS of the above. ... > fact is obvious," then why the strange footnotes in Hardy and Wright? ...
    (sci.math)
Quantcast