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Math Help - Square free numbers

  1. #1
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    Representing as sum of two squares

    Hi, how can one find the number of ways in which an integer can be expressed as a sum of two squares?
    E.g for 5 it's 1, as 5 can be represented as : 1^2+2^2.
    Here a^a+b^b is same as b^b+a^a.
    I know that for an integer to be expressed as such , its prime factorisation should contain no odd powers of primes of the form 4k + 3 . So how do i extend this idea to arrive at the conclusion or is there any other way/formula?
    Thanks.
    Last edited by pranay; March 12th 2011 at 05:17 AM. Reason: sorry, wrong title
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  2. #2
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    Quote Originally Posted by pranay View Post
    Hi, how can one find the number of ways in which an integer can be expressed as a sum of two squares?
    E.g for 5 it's 1, as 5 can be represented as : 1^2+2^2.
    Here a^a+b^b is same as b^b+a^a.
    I know that for an integer to be expressed as such , its prime factorisation should contain no odd powers of primes of the form 4k + 3 . So how do i extend this idea to arrive at the conclusion or is there any other way/formula?
    See here for a relevant discussion.
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  3. #3
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    Thank.
    I worked out the problem, as:
    factorise the number and count the number of factors which are such that factor%4=3 (call these types d1) and factor%4=1 (call these types as d2).
    Then the number of ways of representing the number would be |d1-d3|/2 .
    Now this may take some time for factoring large numbers.So is there any other method to get d1(i.e no of factors with property : factor%4=1) and d3(i.e no of factors with property : factor%4=3) for a number?
    Thanks .
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