Find the smallest integer n that makes 11 x 14^n + 1, a prime number, or, prove that it doesn't exist.

(In reply to re: incomplete solu - thinking better by pcbouhid)

Induction doesn't always have to proceed by showing that if a proposition is true for n then it is true for n + 1.  In this case, given that we know the divisibility is different for even and odd n, we note that:

f(0) = 12 is divisible by 3,
f(1) = 155 is divisible by 5.

Then f(n+2) - f(n) = 11 × 14ⁿ(14² - 1) = 11 × 3 × 5 × 13 × 14ⁿ is divisible by both 3 and 5, and the result follows by induction.
(f(n) divisible by 3 implies f(n+2) divisible by 3; f(n) divisible by 5 implies f(n+2) divisible by 5.)

Posted by Nick Hobson on 2005-06-22 11:36:15