#
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!

postscript (Cambridge UK).

postscript (Canada mirror).

pdf (Cambridge UK).

pdf (Canada mirror).

All postscript files are compressed with gzip -
see this page for advice about gzip, if needed.

related publications.

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

Canadian mirrors:
home page,
publications.
bibtex file.