## Saturday, May 06, 2017

### [euzimeht] a^n \pm b^n

Consider numbers of the form the sum or difference of two powers (same exponent): a^n + b^n or a^n - b^n.  Integers of that form, or integer sequences indexed by n, seem interesting.  They could use a name.

Famously, Fermat's Last Theorem.

The Cunningham Project investigates b==1.  The above expression has algebraic factorizations.  Inspired by "The search for Aurifeuillian-like factorizations" and the reference to Schinzel.  We could imagine a much larger project investigating b not equal to 1.

Fibonacci numbers are almost this form, where a and b are not integers.  We need to divide through by sqrt(5).  Fibonacci also have neat algebraic properties.  We could define sequences​ r*(a^n - b^n) permitting real (maybe even complex) r a b, such that the value is integer for all non-negative integer n.