Using Bonse's Inequality to Find Upper Bounds on Prime Gaps

2 The relationship log(Γ(p k )) has <strong>to</strong> the upper boundp 1120 −δkand other conclusions.Table 1 suggests that our formula for Γ(p k ) indeed is an upper bound for g(p k ), as k getslarge. Also from Table 1 one can see that the fast rate of increase of Γ(p k ) by far exceeds theslower rate of increase of (log(p k )) 2 . Hence it could be interesting <strong>to</strong> investigate by computerthe behavior of the three respective functions Γ(p k ), g(p k ) and (log(p k )) 2 , where (log(p k )) 2by Cramér’s conjecture is such that for large k,gaps.Rankin [2] showed that there exists a real constant c such thatp k+1 −p k(log(p k )) 2 = O(1) is true for maximal primec(log k)(log log k)(log log log log k)(log log log k) 2 < p k+1 − p k . (6)Combined with the result from Theorem 3, this indicates that for large integer k ≫ 4, takenlarge enough so that p k and p k+1 both are very large, each term of the prime differencesequence ∆p k is bounded, for each such k, asc(log k)(log log k)(log log log log k)(log log log k) 2< p k+1 − p k< p k( ∏ k−1i=1 p i − p k )log((k + 1) k+1 k k ) = Γ(p k) + 1.Let, for some real constant c and for any fixed integer k = k 0 ≫ 4,a(k 0 ) = c log(k 0) log log(k 0 ) log log log log(k 0 )(log log log(k 0 )) 2 − 1, (7)b(k 0 ) = p k 0( ∏ k 0 −1i=1 p i − p k0 )log((k 0 + 1) k 0+1k k 00 ) − 1 = Γ(p k 0). (8)Then it follows from Theorem 3 that, for each such positive integer k = k 0 ≫ 4, the valuefor the prime gap g(p k0 ) lies inside an open interval (a(k 0 ),b(k 0 )) on the real line.Remarks: The upper bound Γ(p k ) found in Section 1 and which appears on the real linein Eq. 8 as b(k 0 ) for any large fixed integer k 0 ≫ 4 such that g(p k0 ) ∈ (a(k 0 ),b(k 0 )) ⊆ R 1 ,admittedly, is a large upper bound. This actually can be an advantage. We now can comparethe values of g(p k ) <strong>to</strong> those for log(Γ(p k ), log log(Γ(p k )) and log log log(Γ(p k )). In fact aninspection of Tables 2 and 3 shows it might be profitable <strong>to</strong> compare the values of log(Γ(p k )),log log(Γ(p k )) and log log log(Γ(p k )) <strong>to</strong> those for g(p k ), ∆p k and (log(p k )) 2 , whenever k > 4is large enough so that both p k and p k+1 are two very large consecutive primes. This isbecause one should be able <strong>to</strong> find by computer that log(Γ(p k )) > g(p k ), log(Γ(p k )) > ∆p kand log(Γ(p k )) > (log(p k )) 2 as k grows large, after which one even might be able <strong>to</strong> prove4