# Utomo Theorem ~ The Next Millenium Prize Problem

Utomo Theorem For every prime numbers p which p + 2 is also prime, then $$2^{p+2}$$ - 1 always prime.

Can you proves this conjecture?

Note by Budi Utomo
1 year, 5 months ago

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What you mean is the larger one of every twin prime would produce a Mersenne Prime ? Oh I need to check this out

- 1 year, 4 months ago

It's actually easier to list the primes $$p$$ which satisfy your statement than to find the list that fails it. Just compare any list of twin primes with a list of Mersenne primes, and you'll see that the $$p$$ which satisfy your statement are $p = 3, 5, 11, 17, 29, 59, 1277, 4421, 110501, 132047, \ldots$ and at this point, I got a little tired of looking.

Point being, in the first 8000 or so twin primes, only these 10 satisfy your statement, so there's no way to patch it by removing just a few errant counterexamples.

- 1 year, 5 months ago

you sure that 2^43 - 1 isn't prime number :(

- 1 year, 5 months ago

but, in f(p) = 2^p - 1 hasn't always gives a prime marsenne.

- 1 year, 5 months ago

maybe, except just for p = 41.

- 1 year, 5 months ago

From this theorem, we knew that a large primes is infinite.

- 1 year, 5 months ago

For a counter example, take p=41, 2^43-1 is not prime. See this

- 1 year, 5 months ago

Actually, the numbers of the form 2^n-1 that are prime are known as Mersenne primes. Read this

- 1 year, 5 months ago