一條數學MI問題

Let P(n) be the statement,

that 23^n + 1is a multiple of 3n+1

When n = 1

L.H.S. = 23^(1) + 1 = 9 = 31+1

P(1) is true.

Assume P(k) is true for n = k

23^k + 1 = 3k+1N, where N is an integer

When n = k + 1

23^(k+1) + 1 = (23^k)3 + 1

= (3k+1N - 1)3 + 1

= (3k+1N)3 - 3(3k+1N)2 + 3(3k+1N) - 1 + 1

= 3k+2(32k+1N3 - 3k+1N2 + N), which is a multiple of 3k+2

By the Principle of M.I., P(n) is true for all positive integers n.