On The Prime Numbers In Intervals
Kyle D. Balliet

TL;DR
This paper investigates the distribution of prime numbers within specific intervals, generalizing Bertrand's postulate, exploring Legendre's conjecture, and providing explicit bounds and counts for primes in various ranges using elementary and analytic methods.
Contribution
It extends known prime distribution results to broader intervals, offers new bounds related to Legendre's conjecture, and quantifies the number of primes in these intervals with explicit lower bounds.
Findings
Proved existence of primes in intervals between kn and (k+1)n for 2 ≤ k ≤ 8 using elementary methods.
Established that for 2 ≤ k ≤ 519, there is always a prime between kn and (k+1)n for n ≥ k.
Showed a prime exists between n^2 and (n+1)^{2.000001} for all n ≥ 1.
Abstract
Bertrand's postulate establishes that for all positive integers there exists a prime number between and . We consider a generalization of this theorem as: for integers is there a prime number between and ? We use elementary methods of binomial coefficients and the Chebyshev functions to establish the cases for . We then move to an analytic number theory approach to show that there is a prime number in the interval for at least and . We then consider Legendre's conjecture on the existence of a prime number between and for all integers . To this end, we show that there is always a prime number between and for all . Furthermore, we note that there exists a prime number in the interval for any…
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Taxonomy
TopicsAnalytic Number Theory Research · History and Theory of Mathematics · Advanced Mathematical Identities
