Equal Sums of Like Powers - my dissertation
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 nth
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 = 2315183+
3319543 = 2214243+ 3365883
= 2052923 + 3429523= 1078393+
3627533 = 387873+ 3657573.
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 x3+y3=48988659276962496 in
fact has rank 7, which I believe is as high as is known
for curves of that form.
- 635318657 = 1584+594 = 1334+1344
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 280.
- Randy Ekl discovered that a number writable in two ways
as a sum of two fifth powers exceeds 274, and
one writable in two ways as a sum of two sixth powers
exceeds 289
- 15170835645 = 5173+24683 = 7093+24563
= 17333+21523 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 260
- The only solution to a5+b5+c5+d5=e5
in positive integers less than 32779 is [27, 84, 110, 133]
= 144; Randy Ekl discovered that 140685+62375+50275
= 2205 + 141325, and that's the
only solution of that form in positive integers less than
20000.
- There are 207 solutions to a6+b6+c6=d6+e6+f6
with (a,b,c)=(d,e,f)=1 and the common sum less than 57176.
- There are 526 solutions to a4+b4=c4+d4
in integers less than 220; only 218 of these
are given by the known parameterisations due to Zajta
- There are many solutions to a3+b3 =
c3+d3 where a, b, c, d are taken as
quadratic forms in two variables; if we write [u,v,w] for
the form ux2+vxy+wy2, 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?
Back to Tom's maths page