NOTES. The Primes that Euclid Forgot

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1 NOTES Eite by Sergei Tabachnikov The Primes that Eucli Forgot Paul Pollack an Enrique Treviño Abstract. Let q 2. Suosing that we have efine q j for all ale j ale k, let q k+ be a rime factor of + Q k j q j. As was shown by Eucli over two thousan years ago, q, q 2, q 3,... is then an infinite sequence of istinct rimes. The sequence {q i } is not unique, since there is flexibility in the choice of the rime q k+ iviing + Q k j q j. Mullin suggeste stuying the two sequences forme by () always taking q k+ as small as ossible, an (2) always taking q k+ as large as ossible. For each of these sequences, he aske whether every rime eventually aears. Recently, Booker showe that the secon sequence omits infinitely many rimes. We give a comletely elementary roof of Booker s result, suitable for resentation in a first course in number theory.. INTRODUCTION. The following is one version of Eucli s roof that there are infinitely many rimes. Start with q 2. Suosing that q j has been efine for ale j ale k, continue the sequence by choosing a rime q k+, for which q k+ + ky q j. () Then at the en of the ay, the list q, q 2, q 3,... is an infinite sequence of istinct rime numbers. Of course, the sequence {q i } obtaine in this way is not unique, since the relation () is often satisfie by several choices of the rime q k+. Mullin [4] suggeste two natural ways of isensing with the ambiguity. First, we coul agree that at each ste, we always choose the smallest rime q k+ satisfying (); this leas to the sequence (numbere A in the Online Encycloeia of Integer Sequences, or OEIS [6]) 2, 3, 7, 43, 3, 53, 5, 62267, , 39, 280,, 7, 547,... (2) Alternatively, we might always choose the largest ossible q k+, resulting in the sequence (A in the OEIS) 2, 3, 7, 43, 39, 50207, , , ,... (3) We call (2) an (3) the first an secon Eucli Mullin sequences, resectively. For each of (2) an (3), Mullin raise the question of whether every rime eventually aears. Shanks [5] conjecture on robabilistic grouns (bolstere by comutations of Wagstaff; cf. [7]) that every rime is eventually reache by (2), but essentially nothing about the first Eucli Mullin sequence has been rigorously establishe. The secon Eucli Mullin sequence was investigate by Cox an van er Poorten [2]. j htt://x.oi.org/0.469/amer.math.monthly MSC: Primary A4, Seconary A5 May 204] NOTES 433

2 They showe that all of 5,, 3, 7, 9, 23, 29, 3, 37, 4, an 47 are missing an conjecture that in fact infinitely many rimes fail to aear in (3). The Cox van er Poorten conjecture was very recently confirme by Booker []. Theorem (Booker). The secon Eucli Mullin sequence omits infinitely many rimes. There are two key ingreients in Booker s roof. The first is quaratic recirocity for the Jacobi symbol, which is a stale of many first courses in number theory. In aition to this elementary theorem, Booker also makes use of some fairly intricate results in analytic number theory, secifically work of Burgess from the 960s on uer bouns for short character sums. A simle statement calls out for a simle roof! In this note, we resent a variant of Booker s roof, where all of the analytic number theory is relace by very simleto-rove statements about the istribution of squares an nonsquares moulo a rime. There is a cost for this, certainly; our quantitative bouns are weaker than what follows from Burgess s estimates. However, we believe that given how simle Booker s theorem is to state, there is some value in writing out a roof that is accessible to as wie an auience as ossible. Notation. Throughout the aer, we reserve the letter for a rime variable. We use for the usual Legenre Jacobi symbol. a m 2. PRELIMINARIES ON THE DISTRIBUTION OF SQUARES AND NON- SQUARES MODULO A PRIME. Recall that an integer a not ivisible by is calle a quaratic resiue moulo if the congruence x 2 a (mo ) is solvable an a quaratic nonresiue otherwise. We let `(, ) enote the length of the longest run a +, a + 2,...,a + ` of consecutive quaratic resiues mo, an we let `(, ) enote the longest run of consecutive quaratic nonresiues. If we wish integers congruent to 0 moulo to be allowe in the run, we will write `0 in lace of ` in both cases. In this section, we show that all of `(, ), `(, ), `0(, ), an `0(, ) are smaller than 2. As a relue, we rove an uer boun on the smallest ositive quaratic nonresiue moulo, which we enote by n 2 (). Lemma. Let be an o rime. Then n 2 () < 2 +. Proof. Let n n 2 (). Since < n/ne < + n, the least nonnegative resiue of n/ne moulo lies in the oen interval (0, n). So n/ne is a quaratic resiue moulo. Since n is a quaratic nonresiue, the ratio n/ne /ne is also a nonresiue. So by the minimality of n, it must be that + /n > /ne n. n Hence, n 2 < n 2 n + ale, an so n < Lemma 2. Let ale n < be a quaratic nonresiue moulo. Then `(, ) ale max{/n, n }. Proof. Let ` `(, ), an choose a 2 Z so that all of a +, a + 2,...,a + ` are quaratic resiues moulo. Multilying by n, we obtain a sequence na + n, na + 2n,...,na + `n of quaratic nonresiues moulo, each of which iffers from the 434 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 2

3 revious by n. Suose now that `>/n. In this case, every quaratic resiue moulo can be consiere mo as being walle insie one of the intervals (na + jn, na + ( j + )n) with ale j < /ne, or insie (na +/nen, na + n + ). Thus, any run of quaratic resiues has length boune by n. So either ` ale /n or ` ale n, exactly as claime in the lemma. We can now establish an uer boun on the length of any sequence of consecutive squares moulo. Proosition 3. If is an o rime, then `0(, ) <2. Proof. We first rule out long runs of squares containing a multile of. Suose first that is not a square moulo. Then any such run of squares can be viewe, moulo, as a subset of the interval [0, n 2 ()), an thus has length at most n 2 (). On the other han, if is a square moulo, then such a run can be viewe as a subset of ( n 2 (), n 2 ()), an so has length at most 2n 2 (). Consequently, `0(, ) ale max{2n 2 (), `(, )}. By Lemma, we have 2n 2 () < 2. Thus, it suffices to show that `(, ) < 2. If there is any quaratic nonresiue in the half-oen interval ( 2, 2 ], then this boun on `(, ) follows from Lemma 2. So let us suose otherwise. By Lemma, n 2 () < 2 + < 2, an so n 2 () ale 2. With n : n2 (), each of the integers k 2 n with ale k < is a quaratic nonresiue mo. If we ick k as large as ossible with k 2 n ale 2, then the lack of nonresiues in ( 2, 2 ] imlies that (k + ) 2 n > 2. Subtracting the first inequality from the secon yiels (2k + )n > 3 2 3k 2 n, an thus 2k + > 3k 2. But this inequality is false for each k. This roves that `(, ) <2 an comletes the roof of the roosition. It is easier to rule out long runs of nonsquares mo. Proosition 4. For each o rime, we have `0(, ) <2. Proof. Every nonresiue or multile of can be consiere mo as being walle within the interval ( j 2,(j + ) 2 ), for some ale j < b c, or within the interval (b c 2, + ). The number of integers in an interval of the first kin is 2 j < 2, while the number of integers in (b c 2, + ) is b c 2 < ( ) 2 < 2. Remarks. Much of this section is aate from the charming book of Gelfon an Linnik [3]. Lemma an its roof aear, with trivial changes, as that text s Theorem 9.3., while the roof of Proosition 4 comes from the iscussion at the bottom of. 79. The only novelty is our roof of Proosition 3. Gelfon an Linnik state that result as Theorem 9.3.2, but it seems that their roof is incomlete. May 204] NOTES 435

4 3. PROOF OF THE MAIN THEOREM. Throughout this section, the secon Eucli Mullin sequence is enote q, q 2, q 3,... The main theorem is containe in the following roosition. Proosition 5. Let Q, Q 2,...,Q r be the smallest r rimes omitte from the secon Eucli Mullin sequence, where r 0. Then there is another omitte rime smaller than! 2 ry 2 2 Q i. (4) i Remark. Using the results of Burgess, Booker showe that the exonent 2 in (4) can be relace with any real number larger than ,rovie that 22 e is also relace by a ossibly larger constant. Q r Proof. Let X 2 2 i Q 2 i. Let us suose for the sake of contraiction that every rime ale X excet Q,...,Q r aears in the secon Eucli Mullin sequence. Let be the rime in [2, X] that is last to aear in the sequence {q i }, an say aears as the nth term q n. Then is the largest rime iviing + q q n. Moreover, since each rime smaller than that is not a Q i is one of q,...,q n, the only other ossible rime factors of + q q n are Q,...,Q r. Thus, we must have + q q n Q e Qe 2 2 Qe r r e for some exonents e,...,e r 0 an e. We claim it is ossible to choose a natural number ale X satisfying both of the congruences (mo 4), (mo Q Q r ), (5) as well as. (6) Suose for the moment that this has been rove. Since ale X, an is corime to Q Q r, every rime iviing is among the rimes q,...,q n. So if we write 0 2, where 0 is squarefree, then 0 q q n. Hence, + q q n + q q n + q q n + q q n 0 + q q 2 n. 0 (The very first equality uses quaratic recirocity for the Jacobi symbol.) On the other han, we have Q i Q i for each i, 2,...,r an, so that c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 2

5 + q q n ry! ei i Q i e ry! ei e Q i i, + q q n using in the last ste that + q q n + 2 Q <i<n q i 3 (mo 4). This is a contraiction. It remains to establish the existence of a ale X satisfying (5) an (6). The conitions (5) are satisfie by every integer A (mo M), where A : 2Q Q r an M : 4Q Q r. To obtain (6), we look for a small nonnegative integer k with Mk+A. Equivalently, fixing M 0 satisfying MM 0 (mo ), we seek a nonnegative integer k with k + AM 0 M 0. By the results of section 2, we can fin such a k ale max{`0(, ), `0(, )} < 2. Then the corresoning satisfies 0 < Mk + A < 2M + M < 3M ale 3M X. Since 3M 2Q Q r X, we fin that < X. This comletes the roof. ACKNOWLEDGMENTS. We are grateful to Carl Pomerance an the anonymous referee for their thoughtful suggestions. In articular, the current form of Proosition 3 is ue to the referee; our original result was slightly weaker. We also thank Yuliia Glushchenko for hel with the Russian original of [3]. REFERENCES. A. Booker, On Mullin s secon sequence of rimes, Integers 2A (202), available at htt://www. integers-ejcnt.org/vol2a.html. 2. C. D. Cox, A. J. van er Poorten, On a sequence of rime numbers, J. Austral. Math. Soc. 8 (968) A. O. Gel 0 fon, Yu. V. Linnik, Elementary Methos in the Analytic Theory of Numbers. Pergamon Press, Oxfor, A. A. Mullin, Recursive function theory (a moern look at a Eucliean iea), Bull. Amer. Math. Soc. 69 (963) D. Shanks, Eucli s rimes, Bull. Inst. Combin. Al. (99) N. J. A. Sloane, The On-Line Encycloeia of Integer Sequences, ublishe electronically at htt:// oeis.org. 7. S. S. Wagstaff, Jr., Comuting Eucli s rimes, Bull. Inst. Combin. Al. 8 (993) Deartment of Mathematics, University of Georgia, Athens, GA ollack@uga.eu Deartment of Mathematics an Comuter Science, Lake Forest College, Lake Forest, IL trevino@mx.lakeforest.eu May 204] NOTES 437