Function num_prime::nt_funcs::nth_prime_bounds

pub fn nth_prime_bounds<T: ToPrimitive + FromPrimitive>(
    target: &T
) -> Option<(T, T)>

Returns the estimated inclusive bounds (low, high) of the n-th prime. If the result is larger than maximum of T, None will be returned.

Reference:

• [1] Dusart, Pierre. “Estimates of Some Functions Over Primes without R.H.” arXiv preprint arXiv:1002.0442 (2010).
• [2] Rosser, J. Barkley, and Lowell Schoenfeld. “Approximate formulas for some functions of prime numbers.” Illinois Journal of Mathematics 6.1 (1962): 64-94.
• [3] Dusart, Pierre. “The k th prime is greater than k (ln k+ ln ln k-1) for k≥ 2.” Mathematics of computation (1999): 411-415.
• [4] Axler, Christian. “New Estimates for the nth Prime Number.” Journal of Integer Sequences 22.2 (2019): 3.
• [5] Axler, Christian. Uber die Primzahl-Zählfunktion, die n-te Primzahl und verallgemeinerte Ramanujan-Primzahlen. Diss. PhD thesis, Düsseldorf, 2013.

Note that some of the results might depend on the Riemann Hypothesis. If you find any prime that doesn’t fall in the bound, then it might be a big discovery!