Primes between consecutive powers

Abstract

This paper updates two explicit estimates for primes between consecutive powers. We find at least one prime between n 3 and (n + 1) 3 for all n ≥ exp(exp(32.9)), and at least one prime in (n 296 , (n + 1) 296 ) for all positive n. These results are in part obtained with a explicit version of Goldston's (1983) estimate for the error in the Riemann-von Mangoldt explicit formula.

On the error term in the explicit formula of Riemann--von Mangoldt

“…where G(x, h) = (x + log 1−ω (x + h) + x log 1−ω x, and M , ω, and x M can be taken from Table 5. With (5.1), similar working to that of Section 4 in [CH21] gives a condition for primes between n m and (n + 1) m . The only change we make is setting T = x µ /1.8, for any µ ∈ (0, 1), instead of 3 T = x µ .…”

“…To give a short-interval result for all positive n, we need to consider higher powers. In [CH21] it was shown that there are always primes between consecutive m th powers for m = 180. The proof used a number of estimates relating to the error in the prime number theorem, including an explicit version of Goldston's estimate for ψ(x) [Gol83].…”

“…This is to take into account the higher bound on γ in (5.1). We then arrive at a similar condition to (22) of [CH21], in that we have primes between consecutive m th powers for all x satisfying…”

“…where F (x) and E(x) are defined as in Section 4 of [CH21]. For log x M = 10 5 we can take M = 18.83 with ω = 1, and find that for m = 177, we can take µ = 0.005664 to have (5.2) hold for all log x ≥ 10 5 .…”

“…This is done by making the work of Wolke [Wol83] and Ramaré [Ram16] explicit. We are then able to make small improvements on several number-theoretic results appearing in [CH21], [PT21b] and [Wol83].…”