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.

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