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

Robert J. Betts University of Massachusetts, Lowell One University Avenue Lowell, MA 01854 USA

                    Robert.Betts001@umb.edu
                  robert betts@earthlink.net

Abstract

One can apply Bonse’s inequality to points on the real line to find an upper bound on any prime gap, including both for first occurrences and for maximal prime gaps. However, such a result is neither as fine as the upper bound found by Mozzochi, nor as fine as the lower bound obtained by Rankin for maximal prime gaps. Without deep sieve methods, such as those used by Maier and Pomerance to compute a lower bound for maximal prime gaps, we show one can use Bonse’s inequality to arrive at an upper bound for any given prime gap without intricate derivations for any real constants.

1 Introduction and a new upper bound Γ(p_{k}).

Iwaniec, Pintz, and later Mozzochi [5] found good upper bounds on the difference between two consecutive primes, namely,

p_{k+1} − p_{k} ≪ p_{k}^{θ},

where either θ = \frac{11}{20} − \frac{1}{406} or θ = \frac{11}{20} − \frac{1}{384} [5]. Maier and Pomerance improved a lower bound for prime gaps found previously by Rankin [4]. All of them used deep methods beyond the scope of this paper. Here instead we demonstrate the relevance of Bonse’s inequality to the topic of prime gaps.

We use Bonse’s inequality to find an upper bound on the number g(p_{k}), k = 1, 2, …, of composite integers between two consecutive primes, p_{k} and p_{k+1} > p_{k}. The result can be used to find, for each large integer k ≫ 4, an open interval on ℝ^{1} within which one will find the real value for the prime gap. We also remark in Section 2, that computer work can be done on comparing the rate of growth of the upper bound on g(p_{k}) we found and the rate of growth of (log(p_{k}))^{2} as k → ∞, which by Cramér’s conjecture, is the asymptotic limit for a maximal prime gap whenever k is a very large integer. The sequence of prime gaps is related to the sequence of prime differences [9].

Let

Δp_{k} = p_{k+1} − p_{k}  (1)

denote the prime difference function, and

g(p_{k}) = p_{k+1} − p_{k} − 1  (2)

the prime gap, meaning the number of consecutive composites between the consecutive primes p_{k+1} and p_{k}. We show that for large integer k ≫ 4, such that the consecutive primes p_{k}, p_{k+1} are very large, there exists an infinite sequence {Γ(p_{k})}_{k=1}^{∞}, such that each term

Γ(p_{k}) := \frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{log((k + 1)^{k+1}k^{k})} − 1  (3)

> g(p_{k}),

is an upper bound for each term g(p_{k}) that appears in the infinite sequence {g(p_{k})}_{k=1}^{∞} of prime gaps. However the result is not for a least upper bound on prime gaps g(p_{k}). The values Γ(p_{k}) also are an upper bound on the prime differences Δp_{k}. In addition one should be able to show that each term in the sequence {Γ(p_{k})}_{k=1}^{∞} is an upper bound on each corresponding term (i.e., for each k ≫ 4) in the sequence {log(p_{k})^{2}}_{k=1}^{∞}, where Cramér’s conjecture says that, for large k, K, maximal prime gaps G(p_{K}) — meaning maximal if G(p_{K}) is a prime gap such that G(p_{K}) > g(p_{k}) for all k < K — approach the asymptotic limit (log(p_{k}))^{2}.

The graph for the prime difference function Δp_{k} resembles — at the very least — frequency vs. time graphs in the field of signal processing [8] and Gaussian noise [6]. Our theorem below rests on the foundations of two previously proved propositions, which immediately follow. One can prove Bonse’s inequality as an exercise [3]. Uspensky and Heaslet [7] also discuss Bonse’s inequality.

Proposition 1 (Bonse’s inequality). For k > 4,

p_{k+1}^{2} < ∏_{i=1}^{k} p_{i}.

Proposition 2. Let p_{k} be the kth prime. Then

p_{k} ∼ k log k.

For a proof, see Gioia [1]. Now we are ready to prove a theorem that establishes the upper bound on g(p_{k}) through Bonse’s inequality.

Theorem 3. For k ≫ 4,

g(p_{k}) < Γ(p_{k})
= \frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{log((k + 1)^{k+1}k^{k})} − 1.

Proof. By Bonse’s inequality (Proposition 1), when k ≥ 4,

p_{k}^{2} < p_{1}p_{2} ⋯ p_{k−1}
⟹ p_{k+1}^{2} − p_{k}^{2} < (p_{1}p_{2} ⋯ p_{k}) − p_{k}^{2}
⟹ p_{k+1} − p_{k} < \frac{p_{1} ⋯ p_{k} − p_{k}^{2}}{p_{k+1} + p_{k}}
= \frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{p_{k+1} + p_{k}}.

Now when k ≫ 4 grows ever larger, it is true that p_{k} ∼ k log k while by Proposition 2, it remains true that ε = o(1), where ε = |\frac{p_{k}−k log k}{p_{k}}| (See Table 1, and the proof by Gioia [1] of Proposition 2). Furthermore, when k ≫ 4 grows larger it also follows from Proposition 2 and from the properties of logarithms that

p_{k+1} + p_{k} ∼ (k + 1) log(k + 1) + k log k = log((k + 1)^{k+1}k^{k}),

where as k → ∞,

|\frac{(p_{k+1} + p_{k}) − log((k + 1)^{k+1}k^{k})}{p_{k+1} + p_{k}} | = ε′
→ 0.

Letting 0 ≤ ε′ ≪ 1, that is, so that ε′ ≈ 0, it follows that, as k → ∞,

\frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{p_{k+1} + p_{k}} ∼ \frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{log((k + 1)^{k+1}k^{k})}.  (4)

Then as k → ∞,

p_{k+1} − p_{k} < \frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{p_{k+1} + p_{k}} ∼ \frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{log((k + 1)^{k+1}k^{k})},  (5)

where we have used Propositions 1 and 2. Then it follows that for large integer k ≫ 4 such that both p_{k}, p_{k+1} are very large,

g(p_{k}) + 1 = p_{k+1} − p_{k}
= Δp_{k}
< \frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{log((k + 1)^{k+1}k^{k})},
⟹ g(p_{k}) < Γ(p_{k}) = \frac{p_{k}(∏_{i=1}^{k−1} p_{i} − p_{k})}{log((k + 1)^{k+1}k^{k})} − 1.

2 The relationship log(Γ(p_{k})) has to the upper bound p_{k}^{\frac{11}{20}−δ} and other conclusions.

Table 1 suggests that our formula for Γ(p_{k}) indeed is an upper bound for g(p_{k}), as k gets large. Also from Table 1 one can see that the fast rate of increase of Γ(p_{k}) by far exceeds the slower rate of increase of (log(p_{k}))^{2}. Hence it could be interesting to investigate by computer the behavior of the three respective functions Γ(p_{k}), g(p_{k}) and (log(p_{k}))^{2}, where (log(p_{k}))^{2} by Cramér’s conjecture is such that for large k, \frac{p_{k+1}−p_{k}}{(log(p_{k}))^{2}} = O(1) is true for maximal prime gaps.

Rankin [2] showed that there exists a real constant c such that

\frac{c(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, taken large enough so that p_{k} and p_{k+1} both are very large, each term of the prime difference sequence Δp_{k} is bounded, for each such k, as

\frac{c(log k)(log log k)(log log log log k)}{(log log log k)^{2}} < p_{k+1} − p_{k}
< \frac{p_{k}(∏_{i=1}^{k−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}) = \frac{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}) = \frac{p_{k_{0}}(∏_{i=1}^{k_{0}−1} p_{i} − p_{k_{0}})}{log((k_{0} + 1)^{k_{0}+1}k_{0}^{k_{0}})} − 1 = Γ(p_{k_{0}}).  (8)

Then it follows from Theorem 3 that, for each such positive integer k = k_{0} ≫ 4, the value for the prime gap g(p_{k_{0}}) 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 line in Eq. 8 as b(k_{0}) for any large fixed integer k_{0} ≫ 4 such that g(p_{k_{0}}) ∈ (a(k_{0}), b(k_{0})) ⊆ ℝ^{1}, admittedly, is a large upper bound. This actually can be an advantage. We now can compare the values of g(p_{k}) to those for log(Γ(p_{k}), log log(Γ(p_{k})) and log log log(Γ(p_{k})). In fact an inspection of Tables 2 and 3 shows it might be profitable to compare the values of log(Γ(p_{k})), log log(Γ(p_{k})) and log log log(Γ(p_{k})) to those for g(p_{k}), Δp_{k} and (log(p_{k}))^{2}, whenever k > 4 is large enough so that both p_{k} and p_{k+1} are two very large consecutive primes. This is because one should be able to find by computer that log(Γ(p_{k})) > g(p_{k}), log(Γ(p_{k})) > Δp_{k} and log(Γ(p_{k})) > (log(p_{k}))^{2} as k grows large, after which one even might be able to prove that log(Γ(p_{k})) is a better upper bound on g(p_{k}) than is Γ(p_{k}), as k → ∞. In Table 3 the reader can compare the upper bound found by Iwaniec and Pintz [5],

p^{\frac{11}{20} −δ}  (9)

with Γ(p_{k}), log(Γ(p_{k}) and log log(Γ(p_{k})), where in Table 3 we have allowed δ = \frac{1}{406}. As an alternative one could use the refinement found by Mozzochi, who found that, for real δ = \frac{1}{384},

p_{k+1} − p_{k} ≪ p_{k}^{\frac{11}{20} − \frac{1}{384}} . However a comparison of column 3 and column 5 with column 9 in Table 3 elicits the following question:

Does there exist, as k → ∞, a real constant α > 0, such that

g(p_{k}) = p_{k+1} − p_{k} − 1 < log((log(Γ(p_{k}))^{α}) < p_{k}^{\frac{11}{20} −δ} − 1?

Finally we can use the fact that the field of real numbers on the real line ℝ^{1} is closed for the two binary operations (+, ×), to express the upper bound for prime gaps found by Iwaniec and Pintz [5] for real δ = \frac{1}{406}, by using log(Γ(p_{k})). Let

a_{1}, a_{2}, …,  (10)

be an infinite sequence {a_{k}}_{k=1}^{∞} of real numbers such that

p_{k} = a_{k} log(Γ(p_{k})) = log((Γ(p_{k}))^{a_{k}}).  (11)

Then we have at once

p_{k}^{\frac{11}{20} −δ} = (log((Γ(p_{k}))^{a_{k}}))^{\frac{11}{20} −δ}  (12)
⟹ g(p_{k}) ∈ (a(k), b′(k)) ⊆ ℝ^{1},  (13)

where in Eq. (13) we have replaced a(k_{0}) in Eq. (7) with a(k) and, by utilizing the upper bound result found by Iwaniec and Pintz [5],

b′(k) = p_{k}^{\frac{11}{20} −δ}  (14)

= (log((Γ(p_{k}))^{a_{k}}))^{\frac{11}{20} −δ}.  (15)

In the following Tables 1–3, we have chosen δ = \frac{1}{406}, Iwaniec and Pintz’s [5] result) and relative error (see [1, Theorem 45.3]).

ε = |\frac{p_{k} − k log k}{p_{k}} | → 0.  (16)

k p_{k} g(p_{k}) Δp_{k} (log(p_{k}))^{2} k log k  Γ(p_{k})  ε
3 5  1  2  2.59 … 3.27 … −0.44 … 0.34 …
4 7  3  4  3.79 … 5.56 … 10.85 … 0.21 …
5 11 1  2  5.75 … 8.00 … 115.45 … 0.27 …
6 13 3  4  6.58 … 10.74 … 1224.22 … 0.17 …
7 17 1  2  8.03 … 13.65 … 16860.24 … 0.19 …

Table 1

k p_{k} g(p_{k}) Δp_{k} (log(p_{k}))^{2}  Γ(p_{k})  log(Γ(p_{k})) log log(Γ(p_{k}))
3 5  1  2  2.59 …  −0.44 …  ———– ——————
4 7  3  4  3.79 …  10.85 …  2.38 …  0.87 …
5 11 1  2  5.75 …  115.45 …  4.74 …  1.55 …
6 13 3  4  6.58 … 1224.32 … 7.11 …  1.96 …
7 17 1  2  8.03 … 16860.24 … 9.73 …  2.28 …

Table 2

k p_{k} g(p_{k}) Δp_{k} p_{k}^{\frac{11}{20} −δ}  Γ(p_{k})  log(Γ(p_{k}))  a_{k}  log log(Γ(p_{k}))
3 5  1  2 2.42 … −0.44 …  ———– ———– ——————-
4 7  3  4 2.90 … 10.85 …  2.38 … 2.94 …  0.87 …
5 11 1  2 3.72 … 115.45 …  4.74 … 2.32 …  1.55 …
6 13 3  4 4.07 … 1224.32 … 7.11 … 1.83 …  1.96 …
7 17 1  2 4.72 … 16860.24 … 9.73 … 1.74 …  2.28 …

Table 3

References

[1] A. Gioia, The Theory of Numbers: An Introduction, Markham Publishing Company, 1970.

[2] R. Guy, Unsolved Problems in Number Theory, Second Edition, Springer-Verlag, 1994.

[3] A. M. Kirch, Elementary Number Theory, Intext Educational Publishers, NY, 1974.

[4] H. Maier and C. Pomerance, Unusually large gaps between consecutive primes, Trans. Amer. Math. Soc., 322 (1990), 201–237.

[5] C. J. Mozzochi, On the difference between consecutive primes, J. Number Theory, 24 (1986), 186–187.

[6] T. Nicely, First occurrence prime gaps, http://www.trnicely.net/gaps/gaplist.html.

[7] J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill Book Company, NY, 1939.

[8] http://mathworld.wolfram.com/PrimeDifferenceFunction.html.

[9] N. J. A. Sloane, The On-Line Encyclopedia of Integer Seuqences,

    http://www.research.att.com/~njas/sequences/index.html.

2000 Mathematics Subject Classification: Primary 11B05; Secondary 11B99. Keywords: Bonse’s inequality, Cramér’s conjecture, prime difference function, prime gap.

(Concerned with sequence \underline{A001223}.)

Received January 30 2007; revised version received April 13 2007. Published in Journal of Integer Sequences, April 13 2007.

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