An Elementary Splitting of x^m − y^n − c, with a Divisibility Consequence

For an odd positive integer m and any positive integer n, we write x^m − y^n − 1 as the sum of a term depending only on x, a term depending only on y, and the constant −3, each of the first two being a linear factor times an explicit sum. The identity follows directly from the standard factorisations of x^m + 1 and y^n − 1. We state a generalisation to x^m − y^n − c with arbitrary shifts, and record an elementary divisibility consequence: if x^m − y^n = c in integers with m odd, then every common divisor of x + 1 and y − 1 divides c + 2. The note is elementary; the author has not found this splitting stated elsewhere in the literature. Contact: [email protected]