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Thread: prime numbers

  1. #1
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    prime numbers

    In proving that there are infinitely many primes, one could define a function $\displaystyle f: \mathbb{N} \to \mathbb{N} $ by $\displaystyle f(n) = n!+1 $ and show that (1) $\displaystyle f(n) > n $ for all $\displaystyle n \in \mathbb{N} $, (2) $\displaystyle f(n) $ is either prime or composite and (3) $\displaystyle f(n) $ has prime factors greater than $\displaystyle n $ if $\displaystyle f(n) $ is composite.

    So in general can we find a function $\displaystyle f: \mathbb{N} \to \mathbb{N} $ defined by $\displaystyle f(n) = n! + a $ and choose $\displaystyle a $ accordingly to satisfy the above conditions? Could we define a function $\displaystyle f: \mathbb{N} \to \mathbb{N} $ in another way that satisfies the above condtions?
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  2. #2
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    keep in mind that n! +a where a is between 2 and n
    are always composite numbers
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  3. #3
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    Uggg, just never mind. No helpful thoughts... I'm thinking some more.
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