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: