Yes. There are many of these. Ribenboim (2004) lists many examples on pp. 131 to 155, as does Guy in UPNT A17 (pp. 58-65 in the third edition). See also Prime Formulas on MathWorld.Has any mathematician thought about producing a formula or function which spits out all the prime numbers? i.e 1->2, 2->3, 3->3, 4->5, 5->7, 6->11 etc.
Willans, Wormell, Mináč, and Gandhi all give examples of closed-form solutions.You mean a closed-form solution, I assume? As in, piecewise with a finite number of piecewise parts?
It would not solve the Riemann hypothesis.I think it's safe to assume that nothing of the sort yet exists... if it did, I believe it would answer the Riemann hypothesis. No?
A one or two-variable polynomial can't produce all the primes (and only primes). But a 26-variable one can; Jones, Sato, Wada and Wiens gives an explicit example after Matijasevič showed it was possible.I also believe I read something somewhere about a closed-form polynomial never being able to generate "only" prime numbers.
See:
http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
Yes, that was the result I was referring to. Matijasevič showed that is was possible by showing that recursively enumerable sets are precisely Diophantine sets, and Jones, Sato, Wada and Wiens made the polynomial you mention.I guess it is not a coincident that the number of variables in this diophantine equation is the same as the number of variables in the polynomial that CRGreatHouse refered to.
It could solve it couldn't it? If we have such a bijection [tex]f : \mathbb{N} \to \mathbb{P}[/tex] wouldn't [tex]f^{-1}(p)=\pi(p)[/tex]?It would not solve the Riemann hypothesis.
How would that help?It could solve it couldn't it? If we have such a bijection [tex]f : \mathbb{N} \to \mathbb{P}[/tex] wouldn't [tex]f^{-1}(p)=\pi(p)[/tex]?