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The gaps between sums of two squares

Published online by Cambridge University Press:  23 January 2015

Peter Shiu
Peter Shiu*
Affiliation:
353 Fulwood Road, Sheffield S10 3BQ e-mail:, p.shiu@yahoo.co.uk
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Extract

Problems concerning the set

of numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identity

which shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2 , where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.

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Articles
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The Mathematical Gazette , Volume 97 , Issue 539 , July 2013, pp. 256 - 262
Copyright
Copyright © The Mathematical Association 2013

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