Abstract
For a non-zero integer N, we consider the problem of finding 3 integers (a, b, c) such that (formula presented). We show that the existence of solutions is related to points of infinite order on a family of elliptic curves. We discuss strictly positive solutions and prove the surprising fact that such solutions do not exist for N odd, even though there may exist solutions with one of a, b, c negative. We also show that, where a strictly positive solution does exist, it can be of enormous size (trillions of digits, even in the range we consider).
Original language | English (US) |
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Pages (from-to) | 29-41 |
Number of pages | 13 |
Journal | Annales Mathematicae et Informaticae |
Volume | 43 |
State | Published - Jan 1 2014 |
Keywords
- Cubic representation
- Elliptic curve
- Rational points
ASJC Scopus subject areas
- General Computer Science
- General Mathematics