On continued fractions of the square root of prime numbers Alexandra Ioana Gliga March 17, 2006 Nota Bene: Conjecture 5.2 of the numerical results at the end of this paper was not correctly derived from the Mathematica code written for this investigation. Thus, if you wish to check the validity of the numerical con- jectures please run your own Mathematica code. 1 Introduction This paper presents numerical testings concerning the following conjecture exhibited by Chowla and Chowla in [1]. For any positive integer k there√ exist infinitely many primes P with the continued fraction expansion of P having period k . The conjecture improves the similar already proved results for positive integers and, as a special case, for square free numbers. The validation of this conjecture would prove in the case k = 1 , for example, that there are infinitely many primes of the form m2 + 1 , m ∈ Z . 2 Basic Definitions and Notations Definition 2.1 An expression of the form 1 1 + 1 a2+ a + 1 3 a4+... is called a simple continued fraction.We shall denote it more convenien- tely by the symbol [a1, a2, a3, ..., an, ...]. The terms a1, a2, a3, ... are called the partial quotients of the continued fraction. We will discuss only the cases when the partial quotients are positive integers. 1 Definition 2.2 We denote by pn = [a , a , ..., a ] the n th convergent of the qn 1 2 n simple continued fraction from (1) . Thus, pn and qn are the positive integer numerator and denominator of the n th convergent. Definition 2.3 A continued fraction which is periodic from the first partial quotient is called purely periodic. If the period starts with the second partial quotient, the continued fraction is called simply periodic . We shall denote a simply periodic continued fraction by [a0, a1, a2, ··· , an] Definition 2.4 A quadratic irrational α is said to be reduced if α > 1 is the root of a quadratic equation with integral coefficients whose conjugate root α˜ lies between -1√ and 0. A reduced quadratic irrational associated to D can P + D be written as Q , where P, D, Q are integers, D, Q > 0. Definition 2.5 For a given k ∈ Z+ and a set of positive integers {an}n=0,1,···,k−1 we define P−1 = 1, Q−1 = 0 P0 = a0, Q0 = 1 Pn = anPn−1 + Pn−2, Qn = anQn−1 + Qn−2 for n = 1, 2, ··· , k − 1 3 A Few General Results This section provides√ the necessary background for working with the contin- ued fractions of N, where N∈ Z+. For more similar results, see [4]. Most of the results presented here are encountered there. Theorem 3.1 If α is a reduced quadratic irrational,then the continued frac- tion for α is purely periodic. In order to prove the theorem we need a preliminary lemma: Lemma 3.2 For any given D there is only a finite number of reduced quadratic irrational associated to it. Proof of Lemma√ 3.2 :If α is a reduced quadratic irrational,α ˜ is its conjugate P + D and α = Q , then √ √ P + D P − D α = Q > 1, and − 1 < α˜ = Q < 0 2 (1) 2P The conditions α > 1 andα ˜ > −1 imply that α+α ˜ > 0, or Q > 0, and since Q > 0 we√ conclude that P > 0. Also fromα ˜ < 0 and Q√ > 0 it follows that 0 < P√ < D. The inequality α > 1 implies that P + D > Q and, thus Q < 2 D. Once D is fixed√ there is only√ a finite number of positive integers P and Q such that P < D and Q < 2 D, which proves the assumption. Proof of Theorem 3.1: √ P + D As α is a reduced quadratic irrational it can be uniquely expressed as Q , 1 where P, D, Q are positive integers. We can express α in the form α = a1+ , α1 where a1 is the largest integer less than α, and where √ 1 P1+ D α1 = = > 1 α−a1 Q1 is again a reduced quadratic irrational associated with D. Repeating step by step the process we convert α into a continued fraction such that for every n, α = a + 1 n−1 n αn (2) where α = α0, α1, ... are all the quadratic irrationals associated with D and where a1, a2, .. are the partial quotients of the continued fraction expansion. From the above lemma we have that we must arrive to a reduced quadratic irrational which has occured before, so that αk = αl, for 0 ≤ k < l. As 1 1 αk = ak+1 + = αl = al+1 + , αk+1 αl+1 and since ak+1 and al+1 are the greatest integers less than αk = αl, we conclude that ak+1 = al+1. It then follows that αk+1 = αl+1. Thus we have that from lth partial quotient, the continued fraction for α is periodic. We show next that αk = αl for 0 < k < l implies αk−1 = αl−1, αk−2 = αl−2, . , α0 = αl−k. Letα ˜k =α ˜l be the conjugates of the equal complete quotients αk and αl. Then, it follows that 1 1 βk = − = − = βl α˜k α˜l 3 If k 6= 0, then by taking conjugates in (2), we obtain 1 1 α˜k−1 = ak + andα ˜l−1 = al + α˜k α˜l (3) and thus 1 1 βk = ak + and βl = al + βk−1 βl−1 (4) Since αk−1, αl−1 are reduced, we have that 1 1 0 < −α˜k−1 = < 1 and 0 < −α˜l−1 = < 1 βk−1 βl−1 Thus, ak and al in (4) are the largest integers less than βk, βl, respec- tively;from βk = βl we get that ak = al. Thus, from equation (3) we get that αk−1 = αl−1. Continuing this process we get that αk−2 = αl−2, ··· , α0 = αl−k. As for each αn, an is the greatest integer less than αn, we get that a0 = ak−l, a1 = ak−l+1, ···. Thus the continued fraction of α is purely periodic and we can write α = [a0, a1, ··· , al−k−1]. This completes the proof of the theorem. Corollary 3.3 For any√ N, positive integer which is not a perfect square, the continued fraction of N is simply periodic, more precisely √ N = [a1, a2, a3, ··· , an, 2a1], for some n Proof: √ √ Let a1 be the√ greatest integer less than N. Then √N + a1 > 1 and its conjugate, − N + a1 lies between −1 and 0. Thus, N + a1 is a reduced quadratic irrational with the greatest integer less than it equal to 2a1. We can apply Theorem 3.1: √ N + a1 = [2a1, a2, ··· , an] for some n which is equivalent to √ N + a1 = [2a1, a2, a3, ··· an, 2a1] 4 consequently √ N = [a1, a2, a3, ··· , an, 2a1] where a1 > 0 Theorem 3.4 If for a reduced quadratic integer α = [a1, a2, ··· , an] we de- note by β = [an, an−1, ··· , a1] the continued fraction for α with period re- 1 versed, then − β =α ˜ is the conjugate root of the equation satisfied by α . Proof: pn pn We know that if = [a1, a2, ··· , an] then pn = anpn−1 + pn−2, thus = qn pn−1 1 p2 pn an + pn−1 , for any n. Thus, we get recursively that = [a2, a1],..., = p1 pn−1 pn−2 p˜n qn p˜n−1 [an, an−1, ··· , a1] = . Similarly, we get that = [an, an−1, ··· , a2] = , q˜n qn−1 q˜n−1 where by p˜n and p˜n−1 we understand the nth and the (n − 1)th convergents q˜n q˜n−1 of the continued fraction [an, an−1, ··· , a1]. Since the fractions are already reduced we get that p˜n = pn,p ˜n−1 = qn,q ˜n = pn−1 andq ˜n−1 = qn−1 (5) We also have the recurrences α = αpn+pn−1 and β = βp˜n+˜pn−1 αqn+qn−1 βq˜n+˜qn−1 (6) According to (5) we notice that β = βpn+qn βpn−1+qn−1 1 And, thus from (6) we get that α and − β satisfy the same quadratic equa- 1 tion. We conclude that − β =α ˜ where β = [an, an−1, ··· , a1]. Lemma√ 3.5 Except for the term 2a1 the periodic part of the continued frac- tion of N is symmetrical. 5 Proof: √ √ From the continued fraction of N we get that N−a1 = [0, a2, a3, ··· , an, 2a1]. We can easily get now that 1 √ = [a2, a3, ··· , an, 2a1] N−a1 (7) From Theorem 3.4 we get that 1 √ = [an, an−1, ··· , 2a1] N−a1 (8) √ √ where a1 − N is the conjugate of the reduced quadratic a1 + N. However, we know that the continued fraction expansions are unique. Comparing (7) and (8) we conclude that an = a2, an−1 = a3, ···, a3 = an−1, a2 = an √Thus, except for the term 2a1 the periodic part of the continued fraction of N is symmetrical. We must acknowledge that the set of numbers that have simply periodic, almost symmetrical continued fractions is much larger than the one men- tioned above.We have the more general result presented in [7] that states that the square roots of the rational numbers greater than the unity have the above property. Thus, we must be careful when we analyse simply period continued fractions. We will show however that , except for one observation, Lemma 3.5 is the best description one can get for continued fractions of square roots of positive integers. 4 Main Theoretical Results This section presents a theoretical result obtained in [5].The proof, in its majority, is reproduced ad-literam from [5].The purpose is to introduce the reader to a theoretical result that will be tested numerically in the last part of this report.