I am not sure whether this problem should be in the high school math section or the college math section. (It has been a very long time since I was in either!)

I would like to find a proof, disproof or counter-example for a conjecture I have concerning equations of the form:

(with n being a natural number). (Plotting this equation for some n on a Cartesian co-ordinate plane results in a circle with its center at the origin and a radius of sqrt(n).)

Definitions:
An "integer solution" for an equation (such as the above) is one with all variables (in our case, both x and y) being integers. A "rational solution" is one with all variables (again, both x and y in our case) being rational numbers.

Comment:
Some equations of the above form (such as n = 1, 2, 4, 5, 8, 9 or 10) have integer (and, therefore, rational) solutions and some (such as n = 3, 6 or 7) have no rational (and, thus, no integer) solution at all. (It is easy to find integer solutions for each of the n values in the first list. Also, I have been able to construct SPECIFIC proofs showing that no rational solution exists for each of the values in the second list. However I could not discover a GENERAL proof showing which values of n admit rational solutions and which do not.)

Note that this is NOT the same as the Pythagorean Triples problem, because, in this case, the sum of the squares (x² + y²) need not, itself, be a square.

Conjecture:
If the equation x² + y² = n (for some natural number n) has rational solutions, then it also has integer solutions.

OR (using the contra-conditional version of the above):

If the equation x² + y² = n (for some natural number n) has no integer solution, then it also has no rational solution.

Questions:
Does anyone know an elementary proof that shows whether the above conjecture is true or false? Alternatively, is there a counter example that shows the conjecture is false?

(A counter example would have some n with rational solutions but no integer solution.)

If there is a web reference showing a proof, disproof or counter-example of this statement, could you please give me the URL?

Thanks and regards,

anXZMouse
(which is equivalent to aNonYMouse, unless, of course, the XZ mouse happens to be at the origin ;-).

Both conjectures are true. They are corollaries of the well-known theorem on sums of squares:

A positive integer n is the sum of two integer squares if and only if each prime factor of n of the form 4k+3 occurs to an even power in the prime factorization of n.

Note that the specific cases of n=1, 2,..., 10 that you worked out agree with this theorem. A proof for general n can be found in almost any elementary number theory book.

Now, to prove your first conjecture, assume that n is the sum of two rational squares, i.e., n = (a/b)² + (c/d)² where a, b, c, d are integers with gcd(a,b) = gcd(c,d) = 1.

Clear the fractions to get a² d² + b² c² = n b² d². (*)

Since gcd(a²,b²) = gcd(c²,d²) = 1, equation (*) implies that b² divides d² and d² divides b². Therefore, b² = d².

Hence, equation (*) reduces to a² + c² = n b².

By the theorem above, each prime factor of (n b²) of the form 4k+3 occurs to an even power in the prime factorization of (n b²). But then, each prime factor of n of the form 4k+3 occurs to an even power in the prime factorization of n, and so n can be written as the sum of two integer squares. QED.

The proof the second conjecture follows by noting that if n cannot be written as the sum of two integer squares then neither can (n b² d²) for any integers b and d. Therefore, (*) has no solution.

Mr Toad