A Prime in Every Slice: Beyond Bertrand's Postulate
Abstract
Prime in Every Slice: A Surprising Uniformity Beyond Bertrand’s Postulate A simple yet powerful claim: For any integer ν ≥ 2, partitioning the number line into intervals of length ν up to ν(ν + 1) guarantees at least one prime in every slice. This statement has been verified for ν ≤ 10,000 and implies classical results such as Legendre’s and Oppermann’s conjectures, while also generalizing Bertrand’s Postulate. Could this be the hidden symmetry underlying the distribution of primes?
Full text
A Prime in Every Slice: Beyond Bertrand’s Postulate October 14, 2025 Observation. For every integer ν≥2, each interval ((k−1)ν, kν] for k= 1,2, . . . , ν + 1 appears to contain at least one prime. From Bertrand to Equal Slices Bertrand’s Postulate (1845) guarantees one prime in (n, 2n]. But what if we slice the number line into equal pieces? For example, with ν= 5, the intervals (0,5],(5,10],(10,15],(15,20],(20,25],(25,30] contain the primes 2,3,5|7|11,13 |17,19 |23 |29—at least one in each slice. Connections to Classical Problems •Bertrand’s Postulate: The k= 2 case •Legendre’s Conjecture : Asserts a prime between n2and (n+ 1)2. This is implied by our observation: for ν=n, the interval (n2, n(n+ 1)] = ((n)ν, (n+ 1)ν] is one of the ν+ 1 slices, and since n(n+ 1) <(n+ 1)2, any prime in this slice lies in (n2,(n+ 1)2). •Oppermann’s Conjecture: Contained in the k=νand k=ν+ 1 cases •Firoozbakht’s Conjecture: Would imply our observation, since it bounds prime gaps below νfor large ν Evidence and Invitation •Verified for all ν≤10,000 •Average primes per interval grows like ν 2lnν •Most sparse intervals occur around k≈ν/2 •No counterexamples found despite existence of large prime gaps This surprising uniformity in prime distribution goes far beyond Bertrand’s Postulate. If true, any equal partition of [1,(ν+ 1)ν] guarantees a prime in every slice. Can you find a proof—or perhaps a counterexample? The author has verified this holds for ν≤10,000. 1