PROOF

given the conjecture:

“the sequence of the last digits in the sequences of the powers of the integers reoccur every fourth power and repeat in 10 digit intervals within a given sequence”

a partial proof of this statement may be:

(1) any number abcd……..wxyz may be expressed as a sum (a(x)10^n + b(x)10^n-1 + …… + y(x)10^1 + z(x)10^0)

(2) given (1) any number ab may be expressed as the sum (a(x)10^1 + b(x)10^0)………..thus by the binomial theorem any number (ab)^n…….is such that (ab)^n = (a^n(x)10^1(x)n + a number of cross terms + b^n(x)10^0(x)n)

(3) it is clear thus from the binomial theorem in (2) that the last digit in any number (ab)^n is determined only by the last digit in the original number ab…….that is by “b” itself as multiplied successively in the product (ab)^n 

(4) and any term in the binomial expansion of a number (ab)^n is itself multiplied by a coefficient from Pascal’s triangle in which the leading term a^n and the trailing term b^n are multiplied by a coefficient of just “1”

(5) and thus the generalization of the binomial theorem of (a + b)^n is therefore allowable to a general summation (a + b +…..+ y + z)^n………and in the number (ab……yz)^n only the last digit in the original number “z” contributes to the last digit in the product