a + b = c
Then for every integer and all integers as c we can write,
(a + b)n = cn
To prove “No three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two”,
let b = a + d (d = b − a) and substitute:
cn = an + (a + d)n
Therefore c cannot be an integer since all positive integers for n greater than two are written as Zn = (a + b)n = (a + a + d)n, where Z = a + b = a + a + d and Z represents one of the set of all real integers ℤ.
c2 = (a + b)(a + b) = (e + d)(e − d)the Babylonian “difference of two squares” can be rearranged to the Pythagorean Theorem
c2 = e2 − d2or
e2 = c2 + d2for which an infinite set of integer (Pythagorean) triples can be found.
Mark K Malmros
Chimney Point, VT