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Old 2017-06-28, 12:35   #180
Dr Sardonicus
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Feb 2017

7·733 Posts

Originally Posted by mahbel View Post
It turned out that one 4-sq representation can be transformed into another simply by expanding the squares in the first one and re-arranging them to create new 4-sq representations. It can be shown that for N=7*13=91, the first 4-sq rep (5,5,5,4) can be transformed into any other representation.

And in fact, if one used the expanded squares of a given representation, one can find more factors. Take the example of the first 4-sq rep (5,5,5,4).
So, how does this transform one 4-square representation into another? And where does this process stop, short of all the squares being equal to 0 or 1?

Instead of looking at summands n^2, why not use summands 2^n instead? I absolutely guarantee, if N is odd and composite, and you take

1, 2, 4, ... 2^r, with 2^r < sqrt(N) < 2^(r+1), so r < log(N)/(2*log(2)),

then at least one of the sums of these powers of 2 will be equal to a factor of N! I can even spot you that one of the summands must be equal to 1!

Last fiddled with by Dr Sardonicus on 2017-06-28 at 12:39
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