Raise any integer to any integer power: to find all integers that have integer roots for any integer power n,
and n√(cn) = c. Let c = a + b then
cn = (a + b)n
and it follows
(a + b)n
also finds all integers (from integer binomial pairs of a and b) having an integer root for a power n (as for cn).
Given the above equality, then the expression
an + (a + b)n
equates to an integer that does not have an integer root of the integer power n n > 2 (where the n = 2 case is demonstrated here).
(a + b)n − anequates to an integer that does not have an integer root of the integer power n > 2. For the case n = 2 this expression is the difference of two squares identity.
For all positive integers a, b and n, ((a + b)n)(1)/(n) ∈ ℤ It follows that for n greater than two (an + (a + b)n)(1)/(n)∉ℤ.