mathschallenge.net

Problem

Prove that 14ⁿ + 11 is never prime.

Solution

In problems of this nature it is often helpful to substitute values for n to see if anything useful can be pertained. Thus the first seven terms are 25, 207, 2755, 38427, 537835, 7529547, respectively. Although we cannot be certain, it seems when n is odd, 14ⁿ + 11 is divisible by 5, and, although not entirely obvious, when n is even it is divisible by 3.

Let us consider n being even: 14^2k = 196^k. As 196 1 mod 3, it follows that 196^k 1 mod 3. Therefore 14^2k + 11 is divisible by 3.

When n is odd: 14^2k+1 = 1414^2k = 14196^k. As 196 1 mod 5, it follows that 196^k 1 mod 5, and 14196^k 14 4 mod 5. Therefore 14^2k+1 + 11 is divisible by 5.

Hence 14ⁿ + 11 is divisible by 5 and 3 alternately, and can never be prime.

Of course, with this insight we can approach it far more efficiently by noting that 14 -1 mod 15, hence 14ⁿ will be alternately -1/1.

When n is odd: 14ⁿ + 11 -1 + 11 = 10 mod 15 divisible by 5
When n is even: 14ⁿ + 11 1 + 11 = 12 mod 15 divisible by 3

Prove that (14ⁿ + 4)/2 is never prime.