A Variation on Mills-Like Prime-Representing Functions

László Tóth Rue des Tanneurs 7 L-6790 Grevenmacher Grand Duchy of Luxembourg

                   uk.laszlo.toth@gmail.com

Abstract

Mills showed that there exists a constant A such that ⌊A^{3^{n}}⌋ is prime for every positive integer n. Kuipers and Ansari generalized this result to ⌊A^{c^{n}}⌋ where c ∈ ℝ and c ≥ 2.106. The main contribution of this paper is a proof that the function ⌈B^{c^{n}}⌉ is also a prime-representing function, where ⌈X⌉ denotes the ceiling or least integer function. Moreover, the first 10 primes in the sequence generated in the case c = 3 are calculated. Lastly, the value of B is approximated to the first 5500 digits and is shown to begin with 1.2405547052 . . ..

1 Introduction

Mills [6] showed in 1947 that there exists a constant A such that ⌊A^{3^{n}}⌋ is prime for all positive integers n. Kuipers [5] and Ansari [1] generalized this result to all ⌊A^{c^{n}}⌋ where c ∈ ℝ, c ≥ 2.106, i.e., there exist infinitely many A’s such that the above expression yields a prime for all positive integers n. Caldwell and Cheng [2] calculated the minimum constant A for the case c = 3 up to the first 6850 digits (\underline{A051021}), and found it to be approximately equal to 1.3063778838 . . .. This process involved computing the first 10 primes b_{i} in the sequence generated by the function (\underline{A051254}), with b_{10} having 6854 decimal digits.

The main contribution of this paper is a proof that the function ⌈B^{c^{n}}⌉ satisfies the same criteria, where ⌈X⌉ denotes the ceiling function (the least integer greater than or equal to X). In other words, there exists a constant B such that for all positive integers n, the expression ⌈B^{c^{n}}⌉ yields a prime for c ≥ 3, c ∈ ℕ. Moreover, the sequence of primes generated by such functions is monotonically increasing. Lastly, analogously to [2] the case c = 3 is studied in more detail and the value of B is approximated up to the first 5500 decimal digits by calculating the first 10 primes b_{i} of the sequence.

In contrast to Mills’ formula and given that here the floor function is replaced by a ceiling function, the process of generating the prime number sequence P_{0}, P_{1}, P_{2}, … involves taking the greatest prime smaller than P_{n}^{c} at each step instead of smallest prime greater than P_{n}^{c}, in order to find P_{n+1}. As a consequence, the sequence of primes generated by ⌈B^{c^{n}}⌉ is different than the one generated by ⌊A^{c^{n}}⌋ for the same value of c and the same starting prime (apart from the first element of course).

2 The prime-representing function

This paper begins with a proof of the case c = 3 and will proceed to a generalization of the function to all c ≥ 3, c ∈ ℕ.

By using Ingham’s result [4] on the difference of consecutive primes:

p_{n+1} − p_{n} < Kp_{n}^{5/8},

and analogously to Mills’ reasoning [6], we construct an infinite sequence of primes P_{0}, P_{1}, P_{2}, … such that ∀n ∈ ℕ : (P_{n} − 1)^{3} + 1 < P_{n+1} < P_{n}^{3} using the following lemma.

Lemma 1. ∀N > K^{8} + 1 ∈ ℕ : ∃p ∈ ℙ : (N − 1)^{3} + 1 < p < N^{3}, where ℙ denotes the set of prime numbers.

Proof. Let p_{n} be the greatest prime smaller than (N − 1)^{3}.

(N − 1)^{3} < p_{n+1}
< p_{n} + Kp_{n}^{5/8}
< (N − 1)^{3} + K ((N − 1)^{3})^{5/8}  (since p_{n} < (N − 1)^{3})
< (N − 1)^{3} + (N − 1)^{2}  (since N > K^{8} + 1)
< N^{3} − 2N^{2} + N
< N^{3}.

Note that since (N −1)^{3} < p_{n+1}, (N −1)^{3} +1 < p_{n+1} since (N −1)^{3} +1 = N(N^{2} −3N +3) is not prime.

Given the above we can construct an infinite sequence of primes P_{0}, P_{1}, P_{2}, … such that for every positive integer n, we have: (P_{n} − 1)^{3} + 1 < P_{n+1} < P_{n}^{3}.

We now define the following two functions:

∀n ∈ ℤ^{+} : u_{n} = (P_{n} − 1)^{3^{−n}},
∀n ∈ ℤ^{+} : v_{n} = P_{n}^{3^{−n}}.

The following statements can immediately be deduced:

• u_{n} < v_{n},

• u_{n+1} = (P_{n+1} − 1)^{3^{−n−1}} > ((P_{n} − 1)^{3} + 1) − 1)^{3^{−n−1}} = (P_{n} − 1)^{3−n} = u_{n},

• v_{n+1} = P_{n+1}^{3^{−n−1}} < (P_{n}^{3})^{3^{−n−1}} = P_{n}^{3^{−n}} = v_{n}.

It follows that u_{n} forms a bounded and monotone increasing sequence.

Theorem 2. There exists a positive real constant B such that ⌈B^{3^{n}}⌉ is a prime-representing function for all positive integers n.

Proof. Since u_{n} is bounded and strictly monotone, there exists a number B such that

B := lim_{n→∞} u_{n}.

From the above deduced properties of u_{n} and v_{n}, we have

u_{n} < B < v_{n},
(P_{n} − 1)^{3^{−n}} < B < P_{n}^{3^{−n}},
P_{n} − 1 < B^{3^{n}} < P_{n}.

Theorem 3. There exists a positive real constant B such that ⌈B^{c^{n}}⌉ is a prime-representing function for c ≥ 3, c ∈ ℕ and all positive integers n.

Proof. We can use the generalizations to Mills’ function as shown by Kuipers [5] and Dudley [3] in order to show that ⌈B^{c^{n}}⌉ is also a prime-representing function for c ≥ 3, c ∈ ℕ. This proof is short as it is essentially identical to the one presented above, with the following modifications.

As shown by Kuipers [5] for Mills’ function, we first define a = 3c − 4, b = 3c − 1. Therefore a/b ≥ 5/8. This means that in Ingham’s equation there exists a constant K′ such that

p_{n+1} − p_{n} < K′p_{n}^{a/b}.

Lemma 1 can then be modified by taking N > K^{′b} + 1, defining p_{n} as the greatest prime smaller than (N − 1)^{c} and noticing that ca + 1 = b(c − 1). Analogously to the proof in Lemma 1, we quickly obtain the bounds (N − 1)^{c} + 1 < p < N^{c}. This means that we can construct a sequence of primes P_{0}, P_{1}, P_{2}, … such that for every positive integer n, (P_{n} − 1)^{c} + 1 < P_{n+1} < P_{n}^{c}.

This is then concluded with a similar reasoning as in the proof of Theorem 2.

3 Numerical calculation of B

In this section, a numerical approximation of B is presented for the case c = 3. Mills [6] suggested using the lower bound K = 8 for the first prime in the classical Mills function ⌊A^{3^{n}}⌋, where K is the constant defined in Ingham’s paper [4]. Other authors, including Caldwell and Cheng [2], decided to begin with the prime 2 and then choose the least possible prime at each step. In this case, since the ceiling function replaces the floor function, we choose the greatest possible prime smaller than P_{n}^{3} as the next element P_{n+1}.

If p_{i} denotes the i^{th} prime in the sequence, we obtain

• p_{1} = 2

• p_{2} = 7

• p_{3} = 337

• p_{4} = 38272739

• p_{5} = 56062005704198360319209

• p_{6} = 17619999581432728735667120910458586439705503907211069\ 6028654438846269

• p_{7} = 54703823381492990628407924713718713957740513297193414\ 21259587335767096542227048457036456872683352033529421007878\ 29141860830768725102385452609882503551811073140339908096068\ 8125590506176016285837338837682469

The primes p_{8}, p_{9} and p_{10} are far too large to show in this paper — for instance p_{10} has 5528 decimal digits. The primes p_{i} for i ≤ 8 were verified using a deterministic primality test in Wolfram Mathematica 11 with the ProvablePrimeQ function in the PrimalityProving package, while p_{9} and p_{10} were certified prime by the Primo software [7]. The certification of p_{10} took 14 hours and 23 minutes on an Intel i7-4770 CPU and 4GB RAM. The prime certificates for p_{9} and p_{10} as well as the primes themselves can be found alongside this paper as auxiliary files.

The value of B was calculated up to its first 5500 decimal digits. The first 600 are presented below:

\begin{matrix}1.2405547052 & 5201424067 & 4695153379 & 0034521235 & 3396725255 \\ 9232034386 & 1886622104 & 9111642316 & 9209174137 & 7064313608 \\ 3109555650 & 9480848158 & 9481662421 & 8378961303 & 7426392535 \\ 6658242301 & 8524802142 & 1960037621 & 1464734105 & 8229918628 \\ 4182439221 & 9437396337 & 9442594273 & 8936874985 & 9158491115 \\ 7886891108 & 4262398559 & 2731605607 & 5719554304 & 2915944781 \\ 6278755834 & 4774412491 & 8125993063 & 4590081972 & 8945860313 \\ 1303247244 & 0907981721 & 7119324606 & 1009855753 & 6063847008 \\ 6985820925 & 6038920740 & 0817313213 & 1691077511 & 3322609476 \\ 3239264899 & 5703729933 & 8452155290 & 5152647430 & 8960522935 \\ 3735771869 & 0936560934 & 8000430515 & 4856069064 & 6309177739 \\ 2832001365 & 6550953673 & 1549789328 & 9032942357 & 7708168137 \end{matrix}

References

[1] A. Ansari, On prime representing function, Ganita 2 (1951), 81–82.

[2] C. K. Caldwell and Y. Cheng, Determining Mills’ constant and a note on Honaker’s problem, J. Integer Sequences 8 (2005), Article 05.4.1.

[3] U. Dudley, History of a formula for primes, Amer. Math. Monthly 76 (1969), 23–28.

[4] A. E. Ingham, On the difference between consecutive primes, Q. J. Math. 8 (1937), 255–266.

[5] L. Kuipers, Prime-representing functions, Indag. Math. 12 (1950), 57–58.

[6] W. H. Mills, A prime-representing function, Bull. Amer. Math. Soc. 53 (1947), 604.

[7] Primo software, available at http://www.ellipsa.eu/public/primo/primo.html.

2010 Mathematics Subject Classification: Primary 11A41; Secondary 11Y60, 11Y11. Keywords: prime-representing function, Mills’ constant, prime number sequence.

(Concerned with sequences \underline{A051021} and \underline{A051254}.)

Received June 8 2017; revised versions received September 20 2017; September 26 2017. Published in Journal of Integer Sequences, October 29 2017.

Return to Journal of Integer Sequences home page.