Numbers that are Sums of Squares in Several Ways

David MacKay and Sanjoy Mahajan

Which number comes next?

50, 65, 85, 125, 130, 145, 170, 185, 200, 205...
Hint: it's related to Hardy's taxi, number 1729.

This paper is not highly recommended, since it was written in a state of great ignorance.

Kind correspondents have written to me to answer the question raised in the coda of this paper. Here's the answer courtesy of James Swenson:
By Dirichlet's theorem, there are infinitely many primes of the form p=4n+1. Fermat and Euler showed that all such primes are expressible as sums of two squares. Moreover these expressions are unique up to permutation. This resolves your conjecture: there are infinitely many integers that can only be expressed as the sum of two squares in one way.
Thank you for an interesting puzzle!

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related publications.
David MacKay's: home page, publications. bibtex file.
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