Mathematics.

Cyprian’s Last Theorem

Fermat’s Last Theorem is well known.  It states that an+bn=cn has no solutions in positive integers for n>2.  Fermat claimed to have a proof but no-one has ever been able to find one that uses the methods that would have been available to Fermat.  Fermat's Last Theorem was proved for the first time in 1994, by Andrew Wiles.

Cyprian’s Last Theorem (henceforth CLT) states that the sum of n consecutive nth powers is never equal to the next nth power in the sequence, unless n=2 (32+42=52) or n=3 (33+43+53=63).  It is named after Dom Cyprian Stockford, of Downside Abbey, who first propounded it as a problem.  The theorem has not yet been completely proved; but if Fermat's Last Theorem was called a theorem for 300 years without having a proof, surely Cyprian's can be allowed the same indulgence.

Conjectures have been made, and some theorems proved, about sums of nth powers in general.  CLT states that a particular such equation has no solution for n>3, and because it gives the equation such a restricted form (by demanding consecutiveness) one would expect it to be much easier to prove than many nth-power conjectures.

In fact CLT is a good problem in three ways.  Firstly, elementary methods can make good inroads into it; secondly, different elementary methods (for example, number theory and analysis) combine to attack different special cases, which is always rewarding; and thirdly, some cases remain that are more recalcitrant than others.

CLT may seem too simple to attract serious attention; but as Andrew Wiles himself says, “The definition of a good mathematical problem is the mathematics it generates rather than the problem itself”.  Even if more powerful methods are discovered that yield a general proof for all cases (and there is no certainty that they will be), CLT's unique characteristics would make it an excellent problem for young mathematicians to cut their teeth on.

You can see my latest paper on CLT here (PDF format). It is a work in progress and I apologise for any deficiencies in the layout: this is my first adventure with Lyx and TeX. There are also some blog entries detailing the work in progress. If anyone has anything to contribute, please let me know!

Table of roots

Here is a table of roots x of CLSUM_n(x)=x^n:

n x 1.5+n/log2 1.5+n/log2-x
2
5.0000000000
4.3853900818
-0.6146099182
4
7.3294729050
7.2707801636
-0.0586927414
8
13.0427096010
13.0415603271
-0.0011492739
16
24.5832711700
24.5831206542
-0.0001505158
32
47.6663208740
47.6662413084
-0.0000795656
64
93.8325235590
93.8324826169
-0.0000409421
128
186.1649860040
186.1649652338
-0.0000207702
256
370.8299409280
370.8299304676
-0.0000104604
512
740.1598661850
740.1598609351
-0.0000052499
1024
1478.8197245000
1478.8197218703
-0.0000026297
2048
2956.1394450570
2956.1394437406
-0.0000013164
4096
5910.7788881390
5910.7788874812
-0.0000006578
8192
11820.0577752920
11820.0577749624
-0.0000003296
16384
23638.6155500890
23638.6155499248
-0.0000001642