Thread: Prime Question

For which positive integers n is prime?

I know 1 works. I know it can't be an even n because that would end up even, and thus not prime.

I checked 3, 5, and 7 and they all don't work. I don't know how to prove this as working or not though for bigger odd numbers. I feel like 1 might be the only answer, but not sure how to prove that other numbers do/don't work.

Originally Posted by Janu42

For which positive integers n is prime?

I know 1 works. I know it can't be an even n because that would end up even, and thus not prime.

I checked 3, 5, and 7 and they all don't work. I don't know how to prove this as working or not though for bigger odd numbers. I feel like 1 might be the only answer, but not sure how to prove that other numbers do/don't work.

You can also eliminate non-multiples of 5 when you consider mod 5. This leaves only n that are congruent to 5 mod 10.

Not sure how to deal with those.. but the first 200 of them are composite.

Maybe you can factor them. Let f(n) = n^4 + 4^n. There might be a pattern here: (note: f(25) is not given as prime factorization, rather I multiplied some primes to get two factors close together.)

f(5) = 17 * 97
f(15) = 29153 * 36833
f(25) = 33350257 * 33759857
f(35) = 34350564553 * 34368914633

Difference between the two factors shows a pattern:

80 = 2^4 * 5
7680 = 2^9 * 3 * 5
409600 = 2^14 * 5 * 5
18350080 = 2^19 * 5 * 7

Since we're dealing with odd we can factor this way:



And for both factors are greater than one - the left factor is

obvious. As for the second one - for the odds it can be checked

by hand. For we have