1 i<j k (g ih j g j h i ) 0.

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1 CONSECUTIVE PRIMES IN TUPLES WILLIAM D. BANKS, TRISTAN FREIBERG, AND CAROLINE L. TURNAGE-BUTTERBAUGH Abstract. In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple H(x) = {gx+h j } k j=1 of linear forms in Z[x], the set H(n) = {gn+h j } k j=1 contains at least m primes for infinitely many n N. In this note, we deduce that H(n) = {gn+h j } k j=1 contains at least m consecutive primes for infinitely many n N. We answer an old question of Erdős and Turán by producing strings of m+1 consecutive primes whose successive gaps δ 1,..., δ m form an increasing (resp. decreasing) sequence. We also show that such strings exist with δ j 1 δ j for 2 j m. For any coprime integers a and D we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class a mod D. 1. Introduction and statement of results We say that a k-tuple of linear forms in Z[x], denoted by H(x) = {g j x + h j } k j=1, is admissible if the associated polynomial f H (x) = 1 j k (g jx + h j ) has no fixed prime divisor, that is, if the inequality #{n mod p : f H (n) 0 mod p} < p holds for every prime number p. In this note we consider only k-tuples for which (1) g 1,..., g k > 0 and 1 i<j k (g ih j g j h i ) 0. One form of the Prime k-tuple Conjecture asserts that if H(x) is admissible and satisfies (1), then H(n) = {g j n + h j } k j=1 is a k-tuple of primes for infinitely many n N. Recently, Maynard [5] and Tao have made great strides towards proving this form of the Prime k-tuple Conjecture, which rests among the greatest unsolved problems in number theory. The following formulation of their remarkable theorem has been given by Granville [3, Theorem 6.2]. Date: November 8, Mathematics Subject Classification. Primary 11N36; Secondary 11A41. Key words and phrases. consecutive primes in tuples, bounded gaps between primes, Maynard Tao theorem. 1

2 2 W. D. BANKS, T. FREIBERG, AND C. L. TURNAGE-BUTTERBAUGH Theorem (Maynard Tao). For any m N with m 2 there is a number k m, depending only on m, such that the following holds for every integer k k m. If {g j x + h j } k j=1 is admissible and satisfies (1), then {g j n + h j } k j=1 contains m primes for infinitely many n N. In fact, one can take k m to be any number such that k m log k m > e 8m+4. Zhang [10, Theorem 1] was the first to prove that lim inf n (p n+1 p n ) is bounded; he showed that for an admissible k-tuple H(x) = {x + b j } k j=1 there exist infinitely many integers n such that H(n) contains at least two primes, provided that k Zhang s proof was subsequently refined in a polymath project [7, Theorem 2.3] to the point where one could take k 2 = 632 (at least in the case of monic linear forms). Maynard [5, Propositions 4.2, 4.3] has shown that one can take k 2 = 105 and k m = cm 2 e 4m in the Maynard Tao theorem, where c is an absolute (and effective) constant. Another polymath project [8, Theorem 3.2] has since refined Maynard s work so that one can take k 2 = 50 and k m = ce (4 28/157)m. (In [5, 8], only tuples of monic linear forms are treated explicitly, although the results should extend to general linear forms as considered in [3].) The purpose of the present note is to explain some interesting consequences of the Maynard Tao theorem. We refer the reader to the expository article [3] of Granville for the recent history and ideas leading up to this breakthrough result, as well as a discussion of its potential impact. Without doubt, this result and its proof will have numerous applications, many of which have already been given in [3]. We are grateful to Granville for pointing out to us that Corollary 2 (below) can now be proved. The following theorem establishes the existence of m-tuples that infinitely often represent strings of consecutive prime numbers. Theorem 1. Let m, k N with m 2 and k k m, where k m is as in the Maynard Tao theorem. Let b 1,..., b k be distinct integers such that {x + b j } k j=1 is admissible, and let g be any positive integer coprime with b 1 b k. Then, for some subset {h 1,..., h m } {b 1,..., b k }, there are infinitely many n N such that gn + h 1,..., gn + h m are consecutive primes. A special case of Theorem 1, with m = 2, g = 1 (and the weaker bound k ), has already been established in recent work of Pintz [6, Main Theorem], which is based on Zhang s method but uses a different argument to the one presented here. Theorem 1 (which is proved in 2) has various applications to the study of gaps between consecutive primes. To state our results, let us call a sequence

3 CONSECUTIVE PRIMES IN TUPLES 3 (δ j ) m j=1 of positive integers a run of consecutive prime gaps if δ j = d r+j = p r+j+1 p r+j (1 j m) for some natural number r, where p n denotes the n-th smallest prime. The following corollary of Theorem 1 answers an old question of Erdős and Turán[1] (see also Erdős [2] and Guy [4, A11]). Corollary 2. For every m 2 there are infinitely many runs (δ j ) m j=1 of consecutive prime gaps with δ 1 < < δ m, and infinitely many runs with δ 1 > > δ m. Moreover, in the proof (see 2) we construct infinitely many runs (δ j ) m j=1 of consecutive prime gaps with δ δ j 1 < δ j (2 j m), and infinitely many runs with δ j > δ j δ m (1 j m 1). Using a similar argument, we can impose a divisibility requirement amongst gaps between consecutive primes as well. Corollary 3. For every m 2 there are infinitely many runs (δ j ) m j=1 of consecutive prime gaps such that δ j 1 δ j for 2 j m, and infinitely many runs such that δ j+1 δ j for 1 j m 1. In the proof (see 2) we construct infinitely many runs (δ j ) m j=1 of consecutive prime gaps with δ 1 δ j 1 δ j for 2 j m, and infinitely many runs with δ m δ m 1 δ j+1 δ j for 1 j m 1. As another application of Theorem 1, in 2 we prove the following extension of a result of Shiu [9] on consecutive primes in a given congruence class. Corollary 4. Let a and D 3 be coprime integers. For every m 2, there are infinitely many r N such that p r+1 p r+2 p r+m a mod D and p r+m p r+1 DC m, where C m is a constant depending only on m. Shiu [9] attributes to Chowla the conjecture that there are infinitely many pairs of consecutive primes p r and p r+1 with p r p r+1 a mod D (see also [4, A4]), and proved the above result without the constraint p r+m p r+1 DC m.

4 4 W. D. BANKS, T. FREIBERG, AND C. L. TURNAGE-BUTTERBAUGH 2. Proofs Proof of Theorem 1. Replacing each b j with b j + gn for a suitable integer N, we can assume without loss of generality that 1 < b 1 < < b k. Let S be the set of integers t such that 1 t b k, t {b 1,..., b k }. Let {q t : t S} be distinct primes coprime to g such that t b j mod q t for all t S, 1 j k. By the Chinese remainder theorem we can find an integer a such that (2) ga + t 0 mod q t (t S), and therefore (3) ga + b j 0 mod q t (t S, 1 j k). Consider the k-tuple A(x) = {gqx + ga + b j } k j=1 where Q = t S q t. In view of (3) and the fact that gcd(g, b 1 b k ) = 1, we have gcd(gq, ga + b j ) = 1 for each j, and since {x + b j } k j=1 is admissible, it follows that the k-tuple A(x) is also admissible. Moreover, A(x) satisfies (1) (with g j = gq and h j = ga + b j ) as the integers b 1,..., b k are distinct and gq 1. For every N N, the congruences (2) and our choices of Q and a imply that g(qn + a) + t 0 mod q t (t S). Consequently, any prime number in the interval [g(qn + a) + b 1, g(qn + a)+b k ] must lie in A(n). Let m be the largest integer for which there exists a subset {h 1,..., h m } {b 1,..., b k } with the property that the numbers (4) g(qn + a) + h i (1 i m ) are simultaneously prime for infinitely many N N. Since k k m we can apply the Maynard Tao theorem with A(x) to deduce that m m. By the maximal property of m, it must be the case that for all sufficiently large N N, if the numbers in (4) are all prime, then g(qn + a) + b j is composite for every b j {b 1,..., b k } \ {h 1,..., h m }. Hence, for infinitely many N N, the interval [g(qn +a)+b 1, g(qn +a)+b k ] contains precisely m primes, namely, the numbers {gn + h i } m with n = QN + a. Proof of Corollary 2. Let m 2, and let k k m+1. Let A(x) = {x+2 j } k j=1, which is easily seen to be admissible. By Theorem 1, there exists an (m+1)- tuple B(x) = {x + 2 ν j } m+1 j=1 A(x)

5 CONSECUTIVE PRIMES IN TUPLES 5 such that B(n) is an (m + 1)-tuple of consecutive primes for infinitely many n. Here, 1 ν 1 < < ν m+1 k. For such n, writing with some integer r, we have B(n) = {n + 2 ν j } m+1 j=1 = {p r+1,..., p r+m+1 } δ j = d r+j = p r+j+1 p r+j = 2 ν j+1 2 ν j (1 j m). Then j 1 j 1 δ i = (2 ν i+1 2 ν i ) = 2 ν j 2 ν 1 < 2 ν j+1 2 ν j = δ j (2 j m). Hence, δ j 1 δ 1 + +δ j 1 < δ j for each j, which proves the first statement. To obtain runs of consecutive prime gaps with δ j > δ j δ m δ j+1, consider instead the admissible k-tuple {x 2 j } k j=1. This completes the proof. Proof of Corollary 3. Let m 2, and let k k m+1. Put Q = p k p, and define the sequence b 1,..., b k inductively as follows. Let b 1 = 0, b 2 = Q, b 3 = 2Q, and for any j 3 let Note that b j = b j 1 + (b t b s ). 1 s<t j 1 (5) (b u+1 b u ) (b v+1 b v ) (v u 1). Now put A(x) = {x + b j } k j=1, and observe that A(x) is admissible since Q divides each integer b j. By Theorem 1, there exists an (m + 1)-tuple B(x) = {x + b νj } m+1 j=1 A(x) such that B(n) is an (m + 1)-tuple of consecutive primes for infinitely many n. Here, 1 ν 1 < < ν m+1 k. For any such n, writing with some integer r, we have B(n) = {n + b νj } m+1 j=1 = {p r+1,..., p r+m+1 } δ j = d r+j = p r+j+1 p r+j = b νj+1 b νj (1 j m). Then j 1 j 1 δ i = (b νi+1 b νi ) (b t b s ) = b νj +1 b νj 1 s<t ν j

6 6 W. D. BANKS, T. FREIBERG, AND C. L. TURNAGE-BUTTERBAUGH if 2 j m. On the other hand, using (5) we see that ν j+1 1 (b νj +1 b νj ) (b i+1 b i ) = b νj+1 b νj = δ j. i=ν j Hence, δ 1 δ j 1 δ j for 2 j m, which proves the first statement. To obtain runs of consecutive prime gaps with δ m δ m 1 δ j+1 δ j for 1 j m 1, consider instead the admissible k-tuple {x b j } k j=1. The corollary is proved. Proof of Corollary 4. Let m 2, and let k k m. Let {x + a j } k j=1 be any admissible k-tuple with a 1 < < a k, and put b j = Da j + a for 1 j k; then {x + b j } k j=1 is also admissible. Since gcd(d, b j ) = gcd(d, a) = 1 for each j, we can apply Theorem 1 with g = D to conclude that there is a subset {h 1,..., h m } {b 1,..., b k } such that Dn + h 1,..., Dn + h m are consecutive primes for infinitely many n N; as such primes lie in the arithmetic progression a mod D and are contained in an interval of length b k b 1 = D(a k a 1 ), the corollary follows. Acknowledgements. CLT-B is supported by a GAANN fellowship (grant no. P200A90092). In the first draft of this manuscript, we proved Theorem 1 under the assumption that k exp(e 12m ). We thank Andrew Granville for showing that k need not be larger than the number k m in the Maynard Tao theorem and for simplifying our original proof of Theorem 1. We also thank Gergely Harcos, James Maynard, and the referee for providing helpful comments on our earlier drafts. References [1] P. Erdős and P. Turán, On some new questions on the distribution of prime numbers, Bull. Amer. Math. Soc. 54 (1948), [2] P. Erdős, On the difference of consecutive primes, Bull. Amer. Math. Soc. 54 (1948), [3] A. Granville, Primes in intervals of bounded length, Bull. Amer. Math. Soc. (to appear). [4] R. Guy, Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer Verlag, New York, [5] J. Maynard, Small gaps between primes, Ann. of Math. (2) (to appear). [6] J. Pintz, Polignac numbers, conjectures of Erdős on gaps between primes arithmetic progressions in primes, and the bounded gap conjecture, Preprint, arxiv: (2013), 14pp. [7] D. H. J. Polymath, New equidistribution estimates of Zhang type, and bounded gaps between primes, Preprint, arxiv: (2014), 165pp. [8] D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Preprint, arxiv: (2014), 79pp. [9] D. K. L. Shiu, Strings of congruent primes, J. London Math. Soc. (2) 61 (2000),

7 CONSECUTIVE PRIMES IN TUPLES 7 [10] Zhang, Y. Bounded gaps between primes, Ann. of Math. (2) 179 (2014), Department of Mathematics, University of Missouri, Columbia MO, USA. address: bankswd@missouri.edu Department of Mathematics, University of Missouri, Columbia MO, USA. address: freibergt@missouri.edu Department of Mathematics, University of Mississippi, University MS, USA. address: clbutter@olemiss.edu