Sums of Squares, Cubes, and Higher Powers

Sums of Squares, Cubes, and Higher Powers

On Powers as Sums of Two Cubes

In a paper of Kraus, it is proved that x 3 + y 3 = z p for p 17 has only trivial primitive solutions, provided that p satisses a relatively mild and easily tested condition. In this article we prove that the primitive solutions of x 3 + y 3 = z p with p = 4; 5; 7; 11; 13, correspond to rational points on hyperelliptic curves with Jaco-bians of relatively small rank. Consequently, Chabauty metho...

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...

Squares from Sums of Fixed Powers

In this paper, we show that if p and q are positive integers, then the polynomial exponential equation p + q = y2 can have at most two solutions in positive integer x and y. If such solutions exists, we are able to precisely characterize them. Our proof relies upon a result of Darmon and Merel, and Chabauty’s method for finding rational points on curves of higher genus.

On the Waring–goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers

(1.1) p1 + · · ·+ ps = n, where p1, . . . , ps are prime unknowns. It is conjectured that for any pair of integers k, s ∈ N with s ≥ k + 1 there exist a fixed modulus qk,s and a collection Nk,s of congruence classes mod qk,s such that (1.1) is solvable for all sufficiently large n ∈ Nk,s. While a proof of this conjecture appears to be beyond the reach of present methods, some significant progre...