This dissertation (written for the MMath degree at Oxford)
looks at various problems of the form 'in how many ways can an
integer be written as a sum of a given number of positive *n*th
powers'. About the only proven result in this field is that there
exist integers writable in arbitrarily many ways as a sum of two
positive cubes; I give this result with a proof due to Hardy, and
also the results of a large number of searches.

In summary:

- The smallest number writable in five ways as a sum of two
positive cubes is 48988659276962496 = 231518
^{3}+ 331954^{3}= 221424^{3}+ 336588^{3}= 205292^{3}+ 342952^{3}= 107839^{3}+ 362753^{3}= 38787^{3}+ 365757^{3}. This number has I believe now been discovered three times independently, by me in December 1998, by D J Bernstein in 1996, and by David W Wilson in November 1997. After I submitted the dissertation, John Cremona discovered using Ian Connell's apecs package that the associated elliptic curve x^{3}+y^{3}=48988659276962496 in fact has rank 7, which I believe is as high as is known for curves of that form. - 635318657 = 158
^{4}+59^{4}= 133^{4}+134^{4}is the smallest number writable in two ways as a sum of two fourth powers (this was known to Euler); my search gives that, if a number writable in three ways exists, it exceeds 2^{80}. - Randy Ekl discovered that a number writable in two ways
as a sum of two fifth powers exceeds 2
^{74}, and one writable in two ways as a sum of two sixth powers exceeds 2^{89} - 15170835645 = 517
^{3}+2468^{3}= 709^{3}+2456^{3}= 1733^{3}+2152^{3}is the smallest number writable in three ways as a sum of cubes of coprime integers; my search shows that, if one exists writable in four ways, it exceeds 2^{60} - The only solution to a
^{5}+b^{5}+c^{5}+d^{5}=e^{5}in positive integers less than 32779 is [27, 84, 110, 133] = 144; Randy Ekl discovered that 14068^{5}+6237^{5}+5027^{5}= 220^{5}+ 14132^{5}, and that's the only solution of that form in positive integers less than 20000. - There are 207 solutions to a
^{6}+b^{6}+c^{6}=d^{6}+e^{6}+f^{6}with (a,b,c)=(d,e,f)=1 and the common sum less than 5717^{6}. - There are 526 solutions to a
^{4}+b^{4}=c^{4}+d^{4}in integers less than 2^{20}; only 218 of these are given by the known parameterisations due to Zajta - There are many solutions to a
^{3}+b^{3}= c^{3}+d^{3}where a, b, c, d are taken as quadratic forms in two variables; if we write [u,v,w] for the form ux^{2}+vxy+wy^{2}, the simplest solution is [3,-5,-5] + [5,5,-3] = [-4,-4,-6] + [6,4,4]

If you're still interested, why not read the dissertation itself?