sum of two squares????

smslca

If an Even number could be expressed in the form a² + b² . And if there exits two other numbers m,n such that
a² + b² = m² + n²

then , my question is

is there any relation between (a,b) and (m,n) apart from a² + b² = m² + n² ??

hivizman

smslca wrote: »

If an Even number could be expressed in the form a² + b² . And if there exists two other numbers m,n such that
a² + b² = m² + n²

then , my question is

is there any relation between (a,b) and (m,n) apart from a² + b² = m² + n² ??

Is it necessary that the sum is an even number? There are solutions where the sum is an odd number, for example:

625 = 7² + 24²; 625 = 15² + 20²

(These are Pythagorean triplets, as 625 = 25².)

Possibly the smallest value for an even sum is 50, which is equal to 1² + 7² and to 5² + 5². Must a and b, and m and n, which I assume have to be integers, be different?

bnt

There are definitely many even solutions e.g. 650 = 5² + 25² = 11² + 23² = 17² + 19², but as for a relation between them ..?

When the sum of the two squares is a prime number, you have Fermat's theorem on the sum of those squares. However, since you are after even numbers only, that's of no use, since the only even prime number (2) is not a sum of squares. AFAIK, that's that. :cool:

hivizman

bnt wrote: »

The only even prime number (2) is not a sum of squares. AFAIK, that's that.

I know that it took Russell & Whitehead 379 pages in Principia Mathematica to prove that 2 = 1 + 1, but surely that implies that 2 = 1² + 1²?

bnt

hivizman wrote: »

I know that it took Russell & Whitehead 379 pages in Principia Mathematica to prove that 2 = 1 + 1, but surely that implies that 2 = 1² + 1²?

OK, you got me! Long day, and all that ... now what?

hivizman

On the way home from work, I came up with one (not very exciting) result.

Suppose that an even number can be expressed as the sum of two squares of positive integers in at least two distinctive ways. Then, rather obviously, for any one of the sums, both integers must be either odd or even (an odd number squared is odd, and even number squared is even, and an even number is the sum of either two odd or two even numbers).

However, the square of an odd number is equivalent to 1 (mod 4) - that is, the remainder when divided by 4 is 1. On the other hand, the square of an even number is equivalent to zero (mod 4). Hence, if the the sum of two squares is even, then it must be equivalent to either zero or two (mod 4). This implies that, if an even number can be expressed as the sum of two squares in more than one way, then ALL of the sums must consist of either the squares of two odd numbers (in which case the even number is equivalent to two (mod 4)) or the squares of two even numbers (in which case the even number is equivalent to zero (mod 4)).

The example of 650 illustrates this exactly: the three pairs of numbers whose squares sum to 650 - (5, 25), (11,23) and (17, 19) - are all pairs of odd numbers.

MathsManiac

Might be worth playing around with the fact that you can rearrange to get a² - m² = n² - b², (where, without loss of generality, we take a > m >= n >b).

This gives (a-m)(a+m) = (n-b)(n+b), which might lead somewhere.

(It would imply for example, that a-m and a+m cannot both be prime.)

hivizman

MathsManiac wrote: »

Might be worth playing around with the fact that you can rearrange to get a² - m² = n² - b², (where, without loss of generality, we take a > m >= n >b).

This gives (a-m)(a+m) = (n-b)(n+b), which might lead somewhere.

(It would imply for example, that a-m and a+m cannot both be prime.)

In fact, you can also rearrange to get a² - n² = m² - b², so (a-n)(a+n) = (m-b)(m+b).

Given that, for an even sum of squares, a, b, m and n must all be either odd or even, all the factors in the differences of two squares must be even.

Fremen

A question like this can actually lead into some surprisingly deep mathematical waters. This line of inquiry eventually led to the construction of algebraic integers (the Gaussian integers in particular) and the discovery that unique factorisation fails in certain number fields. This in turn led to the invention of ideals and the ideal class group, which motivated much of the development of modern algebra - but that's the tip of the iceberg.

The sum-of-two squares function r(n) tells you how many ways there are to express n as a sum of squares. If you consider, say (-2)^2 + (-3)^2 = 13 and 2^2 + 3^2 = 13 to be the same solution, then r is multiplicative. This property allows you to compute r(n) for any reasonable value of n (say, less than a hundred digits).

This touched on some of my thesis work - there are connections to the geometric distribution and the negative binomial distribution buried somewhere in this circle of ideas.

hivizman

Fremen wrote: »

A question like this can actually lead into some surprisingly deep mathematical waters. This line of inquiry eventually led to the construction of algebraic integers (the Gaussian integers in particular) and the discovery that unique factorisation fails in certain number fields. This in turn led to the invention of ideals and the ideal class group, which motivated much of the development of modern algebra - but that's the tip of the iceberg.

The sum-of-two squares function r(n) tells you how many ways there are to express n as a sum of squares. If you consider, say (-2)^2 + (-3)^2 = 13 and 2^2 + 3^2 = 13 to be the same solution, then r is multiplicative. This property allows you to compute r(n) for any reasonable value of n (say, less than a million digits).

This touched on some of my thesis work - there are connections to the geometric distribution and the negative binomial distribution buried somewhere in this circle of ideas.

Thanks for the information - I suspected that the question had already been analysed, but it's a long time since I studied maths.

hivizman

There's a good discussion of the representation of integers as the sum of 2 or more squares at this website http://mathworld.wolfram.com/SumofSquaresFunction.html.

This gives a formula for calculating the number of distinct representations of a positive integer as the sum of two non-negative integers (thus allowing for the integers to be zero). I won't reproduce the formula, but it implies that the smallest number that can be represented as the sum of two squares in three distinct ways is 325 = 2⁰ x 5² x 13. The three representations are:

325 = 1² + 18² = 6² + 17² = 10² + 15²

We haven't actually answered the original question, which is whether there is any relationship between the pairs of integers other than the fact that the sum of their squares is the same number. My instinctive feeling is that there is no separate relationship (that is, any proposed relationship will be mathematically equivalent to the sum of squares property we start with), but I shall be happy if someone can provide such a relationship.