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Post Closed as "not a real question" by Felipe Voloch, Will Jagy, Franz Lemmermeyer, Daniel Litt, Andrés E. Caicedo
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The New Mersenne conjectureHermite-grand-conjecture implies that f(k)=(2^(2^k+12^5^11^(7k+1))+1)/3 is prime for all natural numbers k$k$.

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?

The New Mersenne conjecture implies that f(k)=(2^(2^k+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?

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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Hermite
Hermite
  • 77
  • 3

Formula with prime-density 1 in the integers

The New Mersenne conjecture implies that f(k)=(2^(2^k+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?