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The Hermite-grand-conjecture implies that f(k)=(2^(2^5^11^(7k+1))+1)/3 is prime for all natural numbers $k$.

Is there any explicit formula that has so far been proven to produce primes for all natural numbers?

If not, is there under some reasonable restriction of closed-form formula a possible non-constructive proof that there exist (or does not exist under even more restrictive conditions) a finite formula that produces primes for all natural integers?

If not, is there any formula that has been proven to output primes with a frequency approaching 1 for input naturals k approaching infinity?

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    $\begingroup$ Could you elaborate on the first line this seems surprising to me. For the rest yes prime generating functions exist, but they tend to be complex. Also, one can show that certain things cannot work, for example a polynomial. See en.wikipedia.org/wiki/Formula_for_primes $\endgroup$
    – user9072
    Commented Oct 8, 2011 at 17:25
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    $\begingroup$ The first line of your posting is nonsense in more than one way. $\endgroup$
    – Franz Lemmermeyer
    Commented Oct 8, 2011 at 17:43
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    $\begingroup$ Please, be more specific. Define "explicit", and define "formula". For example, my formula f(x)=3 returns a prime for all natural numbers x. $\endgroup$
    – Per Alexandersson
    Commented Oct 8, 2011 at 21:37

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It depends a bit on what you accept as "explicit". E.g., there is a positive real number $A$ such that the integer part of $A^{3^n}$ is prime for all positive integers $n$. See Wikipedia on Mills' constant.

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f(3) = 171. The conjecture is false.

edit: This answer was for the original version, which said the "New Mersenne Conjecture" and gave the formula $f(k)=\frac{2^{2^k+1}+1}{3}$. At least that version gives integer results; the new one isn't an integer for any k.

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  • $\begingroup$ To avoid a misconcpention the New Nersenne Conjecture is not the statement, see en.wikipedia.org/wiki/… (So the 'implies' not the 'conjecture' should be the problem.) $\endgroup$
    – user9072
    Commented Oct 8, 2011 at 17:43
  • $\begingroup$ Actually, the New Mersenne Conjecture is not the New Mersenne Conjecture (primes.utm.edu/mersenne/NewMersenneConjecture.html) $\endgroup$
    – Franz Lemmermeyer
    Commented Oct 8, 2011 at 17:45
  • $\begingroup$ @Franz: In the questioners defence. There is not claim this was the NM conjecture ;) $\endgroup$
    – user9072
    Commented Oct 8, 2011 at 17:48

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