prove that isn't prime

Prove that $14^{n}+ 11$ is never prime

#Algebra #Prime

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

I am assuming that you wish this proved for integers $n \ge 1$ , (for $14^{0} + 11 = 12$ is not prime).

If $n$ is odd then the last digit of $14^{n}$ will be $4$ and thus the last digit of $14^{n} + 11$ will be $5$ , implying that $14^{n} + 11$ is divisible by $5$ .

If $n$ is even then let $n = 2k$ for some integer $k$ . Then $14^{n} + 11 = 196^{k} + 11 = (195 + 1)^{k} + 11$ , (A).

Now in the binomial expansion of $(195 + 1)^{k}$ we will have a factor of $195$ in every term except for the term $1^{k} = 1$ . We can then write equation (A) as $1 + 195*m + 11 = 12 + 195*m = 3*(4 + 65*m)$ for some integer $m$ . Thus in this case $14^{n} + 11$ is divisible by $3$ .

So for all integers $n \ge 1$ we can conclude that $14^{n} + 11$ is not prime.

Brian Charlesworth - 6 years, 7 months ago

Great proof! Although I think for the last half, you can look at the equation mod 3, and this get $(-1)^n-1 \pmod{3}$ which for all even n, we get $1-1=0 \pmod{3}$ . It's similar to your method but just a tad simplier.

Trevor Arashiro - 6 years, 6 months ago

Good point. Also, for the first part we could have looked at the expression mod 5, to get $(-1)^{n} + 1 \equiv 0 \mod{5}$ for all odd n.

It takes a bit of intuition to guess at which mod values to 'filter' with, but it makes for a more elegant approach. :)

Brian Charlesworth - 6 years, 6 months ago

14 gives remainder 2 when divided by 3.So,when n=1,2,3,4... , $14^n$ will give remainder 2,1,2,1 .... So every even power gives remainder 1. This means 14^ $n$ +11 will be of the form 3m+12 which cant be a prime. 14 gives remainder 4 when divided by 5.when n=1,2,3,4... $14^n$ will give rem. 4,1,4,1.. So every odd power gives remainder 4. this means $14^n$ +11 will be of the form 5m+15 which cant be prime.

Rutwik Dhongde - 6 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$