Sunday, 6 February 2011

Sums of Squares Problems

Now even my ex-students are trying to hang me up with "Pythagorean-Like" Problems.  Today I got these two in an email. Since the problems date back to Lewis Carroll (whom  my student's assume I must  have gone to school with since I am so very old).

 I knew the problems and the solutions so I will leave them for now as problems for the reader, and post a solution in a day or two; hopefully with a litlte of the history. 
They are suitable for HS level so everyone is invited. Feel free to submit your solutions.

For any two positive integers X and Y, twice the sum of their squares can be expressed as the sum of two squares.  For example 2(42 + 72) =    2(16+49) = 2(65) = 130..but 130 = (112+32)   

And the followup??? Prove that three times the sum of two squares can be expressed as the sum of four squares. 

Enjoy
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1 comment:

  1. Anonymous6 February 2011 at 15:52

    The first result is moderately interesting:
    (a+b)^2 + (a-b)^2 = 2 (a^2 + b^2)

    But the second one is then trivial
    (a+b)^2 + (a-b)^2 + a^2 + b^2 = 3 (a^2 + b^2)

    Do you have some extra constraints for the second problem to make it non-trivial?

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