Function primal::estimate_nth_prime
source · pub fn estimate_nth_prime(n: u64) -> (u64, u64)Expand description
Gives estimated bounds for pn, the nth prime number,
1-indexed (i.e. p1 = 2, p2 = 3).
That is, if (a,b) = estimate_nth_prime(n), a ≤
pn ≤ b. The bounds used are proved in [1], [2,
Théorèmes 1.6–1.8] (both summarised in [2, pp. 14–15]) and [3,
Section 6.1.2].
[1]: Massias, Jean-Pierre; Robin, Guy. “Bornes effectives pour certaines fonctions concernant les nombres premiers.” Journal de théorie des nombres de Bordeaux 8.1 (1996): 215-242.
[2]: Dusart, Pierre. “Autour de la fonction qui compte le nombre de nombres premiers.” PhD diss., Université de Limoges, 1998.
[3]: Dusart, Pierre. “Estimates of Some Functions Over Primes without R.H.” ArXiv:1002.0442. 2010.
§Examples
// the 1e9-th prime
let billionth = 22_801_763_489;
let (low, high) = primal::estimate_nth_prime(1_000_000_000);
// check the bounds are satisfied
assert!(low <= billionth && billionth <= high);