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Never Prime

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: 142k = 196k. As 196 congruent 1 mod 3, it follows that 196k congruent 1 mod 3. Therefore 142k + 11 is divisible by 3.

When n is odd: 142k+1 = 14times142k = 14times196k. As 196 congruent 1 mod 5, it follows that 196k congruent 1 mod 5, and 14times196k congruent 14 congruent 4 mod 5. Therefore 142k+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 congruent -1 mod 15, hence 14n will be alternately -1/1.

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

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

Problem ID: 235 (31 Jul 2005)     Difficulty: 4 Star

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