NEW BOUNDS AND COMPUTATIONS ON PRIME-INDEXED PRIMES

Transcription

1 NEW BOUNDS AND COMPUTATIONS ON PRIME-INDEXED PRIMES Jonathan Bayless Department of Mathematics, Husson University, Bangor, ME, USA Dominic Klyve Department of Mathematics, Central Washington University, Ellensburg, WA, USA Tomás Oliveira e Silva Electronics, Telecommunications, and Informatics Department, Universidade de Aveiro, Aveiro, Portugal tos@ua.pt Received:, Revised:, Accepted:, Published: Abstract In a 009 article, Barnett and Broughan considered the set of prime-inde primes. If the prime numbers are listed in increasing order, 3, 5, 7,, 3, 7,..., then the prime-inde primes are those which occur in a prime-numbered position in the list 3, 5,, 7,.... Barnett and Broughan established a prime-indeed prime number theorem analogous to the standard prime number theorem and gave an asymptotic for the size of the n-th prime-indeed prime. We give eplicit upper and lower bounds for π, the number of prime-indeed primes up to, as well as upper and lower bounds on the n-th prime-indeed prime, all improvements on the bounds from 009. We also prove analogous results for higher iterates of the sequence of primes. We present empirical results on large gaps between prime-inde primes, the sum of inverses of the prime-inde primes, and an analog of Goldbach s conjecture for prime-inde primes.. Introduction Many of the classes of primes typically studied by number theorists concern properties of primes dictated by the positive integers. Twin primes, for eample, are consecutive primes with an absolute difference of. It is interesting, however, to consider a sequence of primes whose members are determined by the primes them-

2 INTEGERS: 3 03 selves. In this work, we consider the set of prime-indeed-primes or PIPs, which are prime numbers whose inde in the increasing list of all primes is itself prime. In particular we have: Definition.. Let P be the sequence of primes written in increasing order {p i } i. The sequence of prime-indeed-primes, or PIPs, is the subsequence of P where the inde i is itself prime. Specifically, the sequence of prime-indeed primes is given by {q i } where q i = p pi for all i. Prime-indeed-primes seem to have been first considered in 965 by Jordan [], who also considered primes indeed by other arithmetic sequences. They were again studied in 975 by Dressler and Parker [], who showed that every positive integer greater than 96 is representable by the sum of distinct PIPs. Later, Sándor [3] built on Jordan s work of considering the general question of primes indeed by an arithmetic sequence by studying reciprocal sums of such primes and limit points of the difference of two consecutive primes. Some of these results can be found in [4, p ]. The direct inspiration of this work is the recent work of Broughan and Barnett [5], who have demonstrated many properties of PIPs, giving bounds on the n-th PIP, a PIP counting function analogous to the prime number theorem, and some results on small gaps between consecutive PIPs. We follow Broughan and Barnett by eplaining PIPs via eample. In the following list of primes up to 09, all those with prime inde the PIPs are in bold type:, 3, 5, 7,, 3, 7, 9, 3, 9, 3, 37, 4, 43, 47, 53, 59, 6, 67, 7, 73, 79, 83, 89, 97, 0, 03, 07, 09. We note also the equivalent definition of PIPs, which may be of some use. If we let, as usual, π be the number of primes not greater than, then an integer p is a PIP if both p and πp are prime.. Preliminary Lemmas In this section we state a few lemmas which will be helpful in the proofs of the theorems in the remaining sections of this paper. The proofs consist of technical details, and contain no insight of direct relevance to the rest of the paper, so we defer their proofs to Section 0. First, we give a pair of bounds comparing combinations of n and log n to. Lemma.. For n 3, log n < + log n.

3 INTEGERS: Lemma.. For n 3, log n log n < + log n + log. n We will also need two lemmas bounding the reciprocal of log π, one for eplicit bounds and one for asymptotic bounds. Lemma.3. For all 33, log + < log π < Lemma.4. For all k, we have log k π = log k log log + + log +. k log log + + O k log and log log π log π = log log + O log. 3. Bounds on PIPs Broughan and Barnett were the first to put upper and lower bounds on q n, the n-th PIP. In particular, they show q n < nlog n + loglog nlog n + loglog n n log n + On loglog n, q n > nlog n + loglog nlog n + loglog n 3n log n + On. By using some theorems of Dusart [6], we show that these bounds can both be improved and made eplicit. Theorem 3.. For all n 3, we have q n > n log n + log n +, and for all n 7, we have q n < n log n + 3 log n + +. log n Proof. We begin with the lower bound. Using the result of Dusart [6] that p n > n log n +

4 INTEGERS: for all n, we have for all n 3 that q n > n log n + log n log n + + log n +. Now, using the fact that we may bound log n log n + n n > log n +, q n > n log n + log n > n log n + log n +. Turning to the upper bound, we use a result of Rosser and Schoenfeld [7], that: p n < n log n + for n 0. To simplify the notation here, let N = n log n This gives that for n 7, Now, Lemma. gives that q n < N log N + log log N. 3. log N < log n + + log n. Also, by Lemma., log log N < + log n + log < + n log n. Putting these into 3., we can bound q n < n log n + which is the bound in the theorem. 3 log n + + log n,

5 INTEGERS: In fact, using an upper bound on p n due to Robin [8], namely p n < n log n for n 70, we are able to prove the somewhat stronger upper bound q n < n log n log n log n for all n Bounds for the Number of Prime-Indeed-Primes Define π to be the number of PIPs not greater than. Note that it follows immediately from the definition that π = ππ. Broughan and Barnett show that π log, and also give the slightly more sophisticated π log log + O log 3. With respect to the classical prime number theorem, a study of the error term has driven much research in number theory. Among the known results [6, page 6] is π = + + log + O log We also have a number of known eplicit bounds on π, including [6, page ] + π +.76, 4. where the lower bound holds for 599 and the upper bound holds for >. Based on this work, we are able to prove the following theorem. Theorem 4.. For all 3, we have π < log while, for all 79, +.5 log.5log + + log, π > log + + log.

6 INTEGERS: Proof. We begin with the upper bound. First, note that i log = i= log log log < for 79. We may bound π with bounds on ππ, using Lemma.3 and 4.3 to see that π π +.76 log π log π < log log.5log + + log π log. Now, to complete the upper bound s proof, we need only show < log π By Lemma.3 and 4.3,.76 log π.76 log.5log + + log <.738, for Then, log < +.5, which establishes 4.4 and thus the upper bound in the theorem for Finally, a computer check verifies the upper bound for For the lower bound, we can use 4. to see that π π log π π log π log + log π + + log π + + log by Lemma.3, for all A computer check shows that the bound also holds for all , completing the proof of the theorem.

7 INTEGERS: Small improvements can be made to the bounds in Theorem 4., at the epense of the relative simplicity in the statement. However, the lower bound is much closer to the truth. Using a stronger version of 4. and a finer version of Lemma.3 it is possible to prove the following theorem. Theorem 4.. π = log log O log log 4 Table gives the values of π and π for powers of 0 up to 0 4. The table also gives values of π 3, the number of PIPs of prime inde up to see a definition and further generalization in Section 7. The values of π up to 0 3, as well as all values of π and of π 3, were computed using an implementation of the Lagarias-Miller-Odlyzko algorithm [9] described in [0]. The value of π 0 4 was computed in 00 by Buethe, Franke, Jost, and Kleinjung [] using a conditional on the Riemann hypothesis analytic method, and latter confirmed in 0 by Platt [] using an unconditional analytic method.. 5. Twin PIPs Twin primes are pairs of primes which are as close together as possible namely, they are consecutive odd numbers which are both prime. We will define twin primeindeed primes, or twin PIPs, with a similar motivation, that is, they are PIPs which are as close together as possible. Since two consecutive indices cannot be prime, there must always be at least one prime between consecutive PIPs ecept for the trivial case q = 3, q 3 = 5. Futhermore, we know that p, p +, and p + 4 cannot be simulatenously prime, since 3 always divides one of them, and thus the smallest distance between consecutive PIPs is 6. To this end, we make the following definition. Definition 5.. Let p and q be PIPs. We say that they are twin PIPs if p q = 6. The twin prime conjecture states that there are infinitely many twin primes, and Broughan and Barnett conjecture that there are infinitely many consecutive PIPs with a difference of 6 and indeed they conjecture that all gaps of even size at least 6 appear infinitely often. However, the Twin Prime Conjecture has a strong form, which states that where π is the number of twin primes not greater than, where dt π C twin log t C twin log, C twin = p 3 0 pp p

8 INTEGERS: Table : Values of π and of π 3 for small powers of 0 n π0 n π 0 n π 3 0 n

9 INTEGERS: is the twin prime constant. One of the reasons that number theorists have faith in the twin prime conjecture is that the strong form of the conjecture seems to be very accurate. Eperimentally we see that the data conform to this conjecture remarkably closely. The twin prime conjecture predicts that the number of primes up to is about d C twin log The third author has calculated this value precisely [3], and found π = , which impressively agree in the first eight digits. The basic idea behind the strong form of the twin prime conjecture is that the events p is prime and p + is prime are not independent events details can be found in, say, [4, pp. 4 6]. Similar reasoning applies to twin PIPs. In particular, we consider the following question: if q is a PIP, what is the probability that q + 6 is also a PIP? We must have either that q, q +, q + 6 are all prime, or that q, q + 4, q + 6 are all prime. But it is not enough for q + 6 to be prime; it must also be a PIP. Combining these ideas, we find the following Theorem 5.. If a prime q with inde n is the first of a pair of twin PIPs, then one of two cases hold. Either: The triple q, q +, q +6 are all prime, or the triple q, q + 4, q + 6 are all prime. Furthermore, the inde of q and q + 6 must each be prime. From this theorem we can construct a heuristic bound on the density of twin PIPs. Conjecture 5.3. The number of twin PIPs up to, π, is asymptotically p 3 p p 3 dt p 5 log 3 tlog t log log t. p>3 Argument. Hardy and Little established conjectures on the density of prime constellations a century ago [5], and though unproven they are widely accepted, and enjoy considerable empirical support. Their conjectured density of either triple given in Theorem 5. is asymptotically P q, q +, q + 6 p p 3 p 3 dt p>3 log 3 t. We epect, as we have throughout this paper, that we can treat the primality of the inde of a prime q as independent of q. Let n be the inde of q. Then n + is

10 INTEGERS: Table : Actual counts, predicted values, absolute and relative errors for π at small powers of 0. i π0 i L0 i L0 i π0 i L0 i π0 i / π 0i the inde of the prime q + 6. The probability of both n and n + being prime is heuristically p> pp p log n = p> pp p log q/ log q. If we simply multiply our earlier heuristic by the probability of this additional restriction, we find an epected density of twin PIPs to be L = p p 3 pp dt p 3 p p>3 log 3 t log t/ log t = p>3 Evaluating the product gives p 3 p p 3 p 5 L dt log 3 tlog t loglog t. dt log 3 tlog t loglog t. This heuristic seems to describe the distribution of twin PIPs quite well. The following table gives the predicted number and actual number of twin PIPs up to various powers of 0, together with the absolute and relative error at each stage.

11 INTEGERS: The Sum of the Reciprocals of the PIPs The asymptotic density of the PIPs is O/ log, from which it follows that the sum of the reciprocals of the PIPs converges as noted first in []. Reciprocal sums have some interest in themselves; bounding Brun s constant, the sum of the reciprocals of the twin primes, has been a goal of many mathematicians since at least 974 [6, 7, 8, 9, 0]. However, the accuracy of bounds on reciprocal sums also measures in an important way how much we understand a particular class of numbers. Bounding a reciprocal sum well requires two things: first, a computationally determined bound on small integers from the class; and second, good eplicit bounds on the density of large integers from the class. By this measure, twin primes are understood much better than, say, amicable numbers. Let B be the sum of the reciprocals of the twin primes. Then B has been shown to satisfy [4].83 < B <.347, and in fact [0].83 < B <.5, assuming the Etended Riemann Hypothesis. By contrast, let P be the Pomerance Constant the sum of the reciprocals of the amicable numbers. Then the best known bounds on P [] are the fairly weak < P < Comparing the size of these intervals shows that, in a measurable way, current mathematical knowledge about twin primes is better than that of amicable numbers. With this is mind, we are interested in determining the accuracy to which we can bound the sum of the reciprocals of the PIPs. By generating all PIPs up to 0 5 using the obvious epedient of employing two segmented Erathosthenes sieves [], one to check the primality of each odd n in that interval and another to check the primality of πn, and by accumulating the sum of the inverses of the PIPs using a 9-bits fractional part, it we find that q q 0 5 When u and v are not PIPs, the sum T u, v = u<q<v v q = dπ π u can be reasonably well approimated by replacing π by li = 0 yields T u, v ˆT u, v = v u d log v log li = log u d log lie. dt log t. This

12 INTEGERS: 3 03 Due to potential arithmetic overflow problems when is large, in this last integral log lie should be evaluated by replacing it by its asymptotic epansion N lie k! + log k ; N = delivers an approimation with relative error close to π e. This is not enough to evaluate ˆT 0 5, directly with an absolute error smaller that 5 0, so ˆT 0 5, 0 00 can be numerically integrated without using the asymptotic epansion, and then ˆT 0 00, can be numerically integrated using the asymptotic epansion. Both Mathematica and pari-gp agree that It follows that k=0 ˆT 0 5, = q q Comparing T k 0 4, k+ 0 4 with ˆT k 0 4, k+ 0 4 for k =,..., 0 suggests that the absolute value of the relative error of the latter is, with high probability, smaller in 0 6. The error of our estimate of q q is therefore epected to be of order 0 8. Another approach to finding the sum of the reciprocals of the PIPs is to use the eplicit upper and lower bounds on π from Theorem 4., our calculations to 0 5, and partial summation to bound the sum. Let us label the upper bound on π as πu, and the lower bound as πl. Then we have q < q π 0 5 π ut q q t dt, 5 and q q > q 0 5 q π πl t 0 t 5 In fact, these functions π u and π l do a fairly good job bounding π past 0 5, as determined by the difference in the integrals above. Numerical calculation with Mathematica gives the following: πl t 0 t dt ; 5 dt. 0 5 π ut t dt Combining these values with the calculation in 6., we can show that the sum of the reciprocals of the PIPs satisfies in good agreement with < q q <.04365,

13 INTEGERS: The Generalized Prime Number Theorem It is natural to consider a further generalization of prime-inde primes. If the set of primes is listed in order, the subsequence of prime-inde primes could be called -primes. Similarly, if the set of -primes is listed in increasing order, we may call the subsequence with prime inde 3-primes. Let a k-prime be a member of the k-th iteration of this process. One may ask for the analogous results on the n-th k-prime and the number of k-primes up to. As noted in Broughan and Barnett [5], it is not hard to establish an analog to the Prime Number Theorem. Namely, defining π k as the number of k-primes less than or equal to, it is easy to show that π k = log k + O log log k In fact, it can be shown that π k Li k as. The proof of this statement is not difficult, but is also not very enlightening. The theorem we prove here is slightly weaker, but shows more of the shape of the three main terms in this asymptotic. Theorem 7.. For all k, π k = k log log k + + k log + O k log k+. Proof. The proof proceeds by induction on k. The case k = is given in 4., so we assume the statement holds up to k. Then, π k+ = π k π = π log k π. k log log π + + log π + O k πlog log π log k+ π by the induction hypothesis. Now, Lemma.4 gives that k log log π k log k + + π log π log π is equivalent to log k k log + + k + O k log log k log π.

14 INTEGERS: Putting this together with 4. gives that π k+ = π k log log k + + k + O k k log = log k+ + + k + completing the theorem s proof. log log log + O k log Using Theorem 4. as the base case and Lemma.3 for the inductive step, we can prove the following theorem giving eplicit bounds on π k. Theorem 7.. For all k, there eists a computable 0 k such that π k < log k +.5 k k.5 log + and for all 0 k. π k > log k + k + k log Note that from the proof of Lemma.3 and Theorem 4. we may choose any 0 k satisfying π k 0 k 3, as π 79 = 3. It is also not difficult to adapt an argument from [5] to prove a proposition on π k for all k. Proposition 7.3. The following inequalities are true for every integer n > and k and for all sufficiently large real numbers, y: a π k n < nπ k, b π k + y π k + k π k y, and c π k + y π k k y log k y. Proof. Each of the statements in this theorem have been shown for k = in [5] and for k = elsewhere a in [3], b and c in [4]. We assume these base cases and follow the argument in [5] to complete the induction for each statement. a From Panaitopol [3], we have πn < nπ for sufficiently large. The induction hypothesis, followed by an application of Panaitopol s result gives for sufficiently large. π k+ n < π nπ k < nπ k+,

15 INTEGERS: b Using Montgomery and Vaughan s [4] bound π+y π+πy together with the induction hypothesis, we have π k+ + y π π k + k πy π k+ + k+ πy for sufficiently large and y. c This follows from part b and Theorem 7.. Note that these bounds are certainly not the best possible. Inequality b in particular seems rather weak. Proving a stronger general theorem, however, seems difficult. 8. Gaps between PIPs Our consideration in Section 5 of twin PIPs, or consecutive PIPs with difference 6, is just a special case of a more general question about gaps between consecutive PIPs. In this section we consider some computational data on other gap sizes. Let qh = min q i+ q i=h q i be the first occurrence of a gap of size h between PIPs or infinity if no such gap eists, let Q; h = q i q i+ q i=h be the number of gaps of h between PIPs up to, and let F h = p> p h p p be the corresponding Hardy-Littlewood correction factor. As epected due to the prime k-tuples conjecture, it was found that the graph of Q0 5 ; h ehibited a rapid oscillation cf., for eample, [5], which disappeared in a graph of Q0 5 ; h/f h. Contrary to what happens with the graphs of smoothed counts of prime gaps i.e., counts divided by F h, the graphs of smoothed counts of PIP gaps first increase, then attain a maimum at an absissa which grows with the count limit, and only then start to decrease eponentially this behavior can be observed in figure of [5]. Table 3 presents the record gaps also known as maimal gaps [6] that were observed up to 0 5. Since a large gap between primes very likely corresponds to a large gap between PIPs having the large prime gap between their indices, the first

16 INTEGERS: ten occurrences of each prime gap up to 4 0 8, obtained as a colateral result of the third author s etensive verification of the Goldbach conjecture [?], were used to locate large gaps between PIPs. This was done as follows:. given an inde i the first prime of a large prime gap, an approimation ˆp i of p i was found by solving ˆπˆp i i < 0, where ˆπ is the Riemann s formula for π, truncated to the first one million comple conjugate zeros on the critical line, and with lower order terms replaced by simpler asymptotic approimations;. using the algorithm described in [0], πˆp i was computed this was by far the most time consuming step; 3. using a segmented sieve and using ˆp i as a starting point, going backwards if necessary, p i, which is by construction a PIP, was located; 4. since the gap between indices was known a priori, the net PIP was also located, and the difference between the two was computed. The maimal gap candidates above 0 5 that resulted from this effort are presented in Table 4; below 0 5, the results of Table 3 were reproduced eactly. Based on the data from these two tables, it appears that h/ log 3 qh is bounded. 9. A Goldbach-like conjecture for PIPs Let Rn be the number of pairs q, n q such that both q and n q are PIPs. Just like for the classical Goldbach conjecture [7], the identity L Rn n = q mod L+, q L n= coupled with a fast polynomial multiplication algorithm based on the Fast Fourier Transform, makes it possible to compute Rn for all n L using only OL +ɛ time and space. For L a positive even integer, let and R lower ; L = min Rn n L R upper = ma n Rn. For n even, these two non-decreasing functions are useful lower and upper bounds of the value of Rn. Figure shows how these two functions behave their points of increase up to L = 0 9 were computed with the help of a simple matlab script. Based on our empirical data, the following conjecture is almost certainly true.

17 INTEGERS: Table 3: Record gaps between PIPs up to 0 5 h qh h qh h qh Table 4: Potential record gaps between PIPs after 0 5 h qh h qh

18 INTEGERS: Number of pairs offset by R lower ; L + R upper Figure : Lower and upper bounds of the number of ways, offset by one, of epressing an even number by an ordered sum of two PIPs. Conjecture 9.. All even integers larger than 806 can be epressed as the sum of two prime-indeed primes. 0. Proofs of lemmas Proof of Lemma.. First, note that From this, log n = log log n Since log + < for > 0, log + n log n = n log n. + which completes the lemma s proof. Proof of Lemma.. First, note that n log n log n = n log n + log n = log + < log n log n, + log n = n log n + +. log n. log n

19 INTEGERS: Now, log + + log n = n because log + < for > 0. Thus, log n log n = log log < log n log n n log n = + log < + < n log n + + log n + log n log n + log n log n + log n, + log n where the last inequality comes from again using log + <. This proves the lemma. Proof of Lemma.3. We begin with the upper bound. From [7], we know that for all 7. Thus, in this range, and so log π π log π < log, log. 0. Writing the second factor as a geometric series proves the bound. Considering the lower bound, we use the upper bound on π for > in 4. to see that log π log +.76 = log + log +.76 log = + log +.76 log < +.76, where the last inequality uses log + < for > 0.

20 INTEGERS: Taking the reciprocal of this inequality, we have log π Thus, to establish the lemma, we need to bound log +.76 log log +.76 log +. log. 0. This is indeed the case, as we may rewrite the fraction as the sum of a geometric series. That is, we may write Now, for 33, k= log +.76 log = + log.76 k log = k= log.76 k log. log.76 log log +.76 log establishing 0.. This completes the lemma s proof. >.76 log, Proof of Lemma.4. Using Lemma.3, together with 4., we have log k π = = log k log log k + + O log k log log + + O k log, 0.3 which establishes the first half of the lemma. We know that π = + + O log, and so the same argument used in the proof of Lemma. gives that log log log log π = log + log + O log log = log + O.

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