Number Cubes with Consecutive Line Sums

Sums of squares, sums of cubes, and modern number theory

These are notes which grew out of a talk for general math graduate students with the aim of starting from the questions “Which numbers are sums of two squares?” and “Which numbers are sums of two cubes?” and going on a tour of many central topics in modern number theory. In the notes, I discuss composition laws, class groups, L-functions, modular forms, and elliptic curves, ending with the Birc...

Perfect Powers That Are Sums of Consecutive Cubes

Euler noted the relation 63= 33+43+53 and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determi...

A Singular K3 Surface Related to Sums of Consecutive Cubes

shows in particular that the sum of the first k consecutive cubes is in fact a square. A lot of related diophantine problems have been studied. For instance, Stroeker [Str] considered the question, for which integers m > 1 one can find non-trivial solutions of m3 + (m + 1)3 + . . + (m + k 1)3 = C2. In particular he found all solutions with 1 < m < 51 and with m = 98. Several authors considered ...

Sums of cubes with shifts

On sums of seven cubes

We show that every integer between 1290741 and 3.375× 1012 is a sum of 5 nonnegative cubes, from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 nonnegative cubes.