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Prove that there is a positive integer that can be written as the sum of squares of positive integers in two different ways. For some reason this is not clicking in my head and I can not figure it out. Any help will be greatly appreciated and I would appreciate no straight forward answer since I am trying to understand.

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Short constructive proof (expands on , and ):

As observed, a single concrete example is a valid existence proof. For a more explanatory approach that produces many examples, use the classical multiplicative identity for sums of two squares:

(a^2 + b^2)(c^2 + d^2)
  = (ac - bd)^2 + (ad + bc)^2
  = (ac + bd)^2 + (ad - bc)^2

This identity shows that the product of two numbers, each expressible as a sum of two squares, itself has two algebraically different representations as a sum of two squares. Pick simple, nonproportional pairs to see it in action. For instance, with (a,b) = (1,2) and (c,d) = (2,3) the product is 65, and the two formulae give the pairs (8,1) and (4,7), so 65 = 1^2 + 8^2 = 4^2 + 7^2 (verify: 1 + 64 = 65 and 16 + 49 = 65). That is one clear example, and the algebra explains why it exists.

Because the identity is multiplicative, it produces infinitely many such integers: vary the starting pairs or multiply more factors of the form (x^2 + y^2). To avoid getting the same representation twice (up to order or sign), choose pairs that are not scalar multiples and avoid zeros. This removes the mystery: the phenomenon is structural, not accidental, and explains the multiple decompositions some members (for example ) pointed out.

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Member Avatar for DavidB

I think one way would be to just provide an example, wouldn't it? For example,

2^2 + 3^2 = 4 + 9 = 13

13 is a positive integer, so you have proved that there is a positive integer that can be expressed as the sum of the squares of positive integers.

In fact, the sum of squares of any integers (x, y, z, etc.) is always a positive integer:

x^2 + y^2 + z^2 + . . . is always going to be a positive integer.

The only way for it not to be a positive integer would be if x, y, z, etc. were non-integer numbers.

Member Avatar for delta_frost

There exist many such examples.But the best example is the Charamichael number 1105.It can be expressed as the sum of squares of two integers in 4 different ways or so.

In general let the statement that the numbers be x,y and z which satisfy:
x^2 + y^2 = z
be true.

If x and y are co-prime,i.e their HCF is 1,one of them is divisible by 3,then the above property satisfies for certain numbers like

13^ + 6^2 = 205 = 3^2 + 14^2
15^2 + 8^2 = 289 = 17^2 + 0^2 (But here zero IS NOT a positive integer)

This is not a generalization;

1105 = 24^2 + 23^2 = 9^2 + 32^2 = 31^2 + 12^2 = 4^2 + 33^2

Edited by delta_frost
Member Avatar for FUTURECompEng

Thanks to both of you I now understand, and from your explanations it seems that I was over complicating the question.

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