Verification of the Firoozbakht conjecture for primes up to four quintillion

Abstract

If p k is the kth prime, the Firoozbakht conjecture states that the sequence (p k ) 1/k is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify the Firoozbakht conjecture for primes up to four quintillion (4 × 10 18 ). Mathematics Subject Classification: 11N05

Verifying the Firoozbakht, Nicholson, and Farhadian Conjectures up to the 81st Maximal Prime Gap

“…While Kourbatov's recent work [26][27][28] yields a useful and explicit domain of validity for the Firoozbakht conjecture, (ultimately, see [29] and [28], for all primes p < 2 64 ), the present article first slightly extends this domain of validity (all primes p < p * 81 ), and second and more significantly obtains identical domains of validity for the related but slightly stronger Nicholson and Farhadian conjectures. The analysis has been presented in such a way that it can now be semi-automated.…”

“…The stronger conjecture that ln pn n is itself monotone decreasing depends on fluctuations in the distribution of the primes p n ; fluctuations which can be rephrased in terms of the prime gaps g n := p n+1 − p n . Indeed, Kourbatov [26] using results on first occurrence prime gaps has recently verified Firoozbakht's conjecture to hold for all primes p < 4×10 18 . Furthermore Kourbatov [27] has also derived a sufficient condition for the Firoozbakht conjecture to hold:…”

“…The Firoozbakht, Nicholson, and Farhadian conjectures would, if proved to be true, impose increasingly strong constraints on the distribution of the primes; this distribution being a fascinating topic that continues to provide many subtle and significant open questions . The Firoozbakht conjecture [24][25][26][27][28] is normally phrased as follows.…”

Strong version of Andrica's conjecture

“…Finally the Cramér conjecture [6] (and closely related conjectures such as the Firoozbakht conjecture [1][2][3][33][34][35][36][37][38]) impose significantly stronger constraints on the prime gaps of the form g n := p n+1 − p n = O (ln p n ) 2 .…”

Bases aditivas de um ponto de vista topológico e prime gaps de um ponto de vista algébrico

“…Um exemplo de um problema satisfazendo as três condições acimaé a conjectura de Firoozbakht. [40]), mas não existe consenso acerca do valor lógico desta conjectura. O autor desta tese, em particular, acredita que elaé verdadeira.…”