[lpetrich]

Let's see what primes have special forms ak+1 and ak-1.

In both of them, if a is odd, then the only prime that they will give us is 2. So I will make a even.

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For the + case, if k has odd factor m, then for b = ak/m, then this expression can be factored (b + 1) * (bm-1 - bm-2 + ... + b2 - b + 1)

That makes this number composite unless k has no odd factors, meaning that k must be 2n. I thus get the generalized Fermat numbers,
F(a,n) = a2^n + 1

For a = 2, one gets the plain Fermat numbers, F(n) = 22^n + 1

The only ones known to be prime are for n = 0, 1, 2, 3, 4 -- 3, 5, 17, 257, 65537. The Fermat numbers for n = 5 to 32 are known to be composite, and it is unknown whether there are any other Fermat primes.

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For the - case, if k has factor m, then for b = ak/m, then this expression can be factored (b - 1) * (bm-1 + bm-2 + ... + b2 + b + 1)

This means that k must be a prime number and that a must be 2. I thus get the Mersenne numbers,
M(n) = 2n - 1
where n being a prime is a necessary condition but not a sufficient one for M(n) also being a prime.

Necessary: M(n) a prime -> n a prime (backward -- true here)
Sufficient: n a prime -> M(n) a prime (forward -- false here)

For primes 2, 3, 5, 7, Mersenne numbers 3, 7, 31, and 127 are also primes, but for 11, we find a composite number: 2047 = 23 * 89. We have been able to find Mersenne primes for as far as we have been able to look, with the current record being 274,207,281 - 1, though they get less and less dense. Currently, 49 Mersenne primes are known, and that record is for the 4,350,601th prime. It is unknown whether there is an infinite number of Mersenne primes.

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Mersenne primes have a connection to perfect numbers: 2n-1*M(n) or M(n)*(M(n)+1)/2. All even perfect numbers have that form, and it is unknown whether there are any odd ones.