Sum of powers of 5 is a prime
5 5 n + 1 + 5 5 n + 1 How many integers n exist such that the above expression is a prime number?
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1 solution
Awesome! Thanks for sharing.
Here is another trick which is quite usual when dealing with primes. It also shows how the problem might have been created originally.
Let m = 5 5 n for some integer m . Then observe that, 5 5 n + 1 + 5 5 n + 1 = m 5 + m + 1 = ( m 2 + m + 1 ) ( m 3 − m 2 + 1 ) . Trivially, both m 2 + m + 1 and m 3 − m 2 + 1 are greater than 1. Hence the conclusion follows.
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Ha, yes, that's neater!! I didn't spot the factorisation.
To get integer results, we only need to consider n ≥ 0 . Trying out a few small values of n (with the help of Wolfram|Alpha) suggests it's worth looking at this expression modulo 3 1 .
Let x n = 5 5 n . We're interested in x n + 1 + x n + 1 .
We have x n + 1 = x n 5 .
From this, it's easy to show that x n ≡ 5 ( m o d 3 1 ) for even n and x n ≡ 2 5 ( m o d 3 1 ) for odd n .
Hence x n + 1 + x n + 1 ≡ 0 ( m o d 3 1 ) for all n , so all of these numbers are divisible by 3 1 , and so none of them are prime. (Trivially, all such integers are greater than 3 1 .)