CATALAN NUMBERS, PRIMES AND TWIN PRIMES

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1 CATALAN NUMBERS, PRIMES AND TWIN PRIMES CHRISTIAN AEBI AND GRANT CAIRNS Abstract. Deser Artkel beabschtgt, enge Bezehungen zwschen Catalan-Zahlen, Prmzahlen und Prmzahlzwllngen darzustellen, ndem nur elementare arthmetsche Begrffe angewendet, glechzetg aber auch vele (hstorsche Bezüge für den an ener weteren Ausführung nteresserten Leser angeboten werden. Das Verhältns zwschen Catalan-Zahlen und Prmzahlen ermöglcht es uns, das Konzept der Catalan-Pseudoprmzahl zu defneren und hre Verbndung mt Weferch-Prmzahlen aufzuwesen. Der zwete Tel enthält Krteren für Prmzahlzwllnge m Zusammenhang mt dem Theorem von Wlson (Clement, dem klenen Theorem von Fermat und Catalan-Zahlen. Der letzte Tel konzentrert sch auf de Untersuchung der zentralen Bnomalkoeffzenten n verschedenen Fällen, so dass Rückschlüsse über de Ncht-Exstenz der Catalan-Pseudoprmzahlen spezfscher Formen gezogen werden können.. INTRODUCTION. Orgnally Catalan numbers were revealed for the frst tme n a letter from Euler to Goldbach n 75 when countng the number of trangulatons of a convex polygon (for a bref hstory see [6]. Today thay are usually defned by C n = ( n+ n n and can be characterzed recursvely by C n = (n n + C n or C n = C n C 0 + C n C + + C C n + C 0 C n, wth C 0 =. Ther appearances occur n a dazzlng varety of combnatoral settngs where they are used to enumerate all manner of geometrc and algebrac objects (see Rchard Stanley s collecton [9, Chap. 6]; an onlne Addendum s contnuously updated. Qute a lot s known about the dvsblty of the Catalan numbers; see [, 0]. They are obvously closely related to the mddle bnomal numbers and not surprsngly, there s a consderable lterature on ther dvsblty too; see [3, 5, 9, 5, 7, 6]. The am of ths paper s to observe a connecton between Catalan numbers, prmes and twn prmes.. PRIMES. There are few better known or more easly understood problems n pure mathematcs than the queston of rapdly determnng whether a gven nteger s prme [8]. A classc prmalty crteron s Wlson s theorem, whch says (see [4, Ch. ]: Wlson s Theorem. A natural number p s prme f and only f (p! (mod p. Wlson s theorem s a very strkng result, and yet t s qute mpractcal as a prmalty check. The trouble wth Wslon s theorem s that t s more beautful than useful [6]. In some texts, t s used to prove Fermat s lttle theorem, a partcular case of whch s: Theorem. If p s prme, then p (mod p.

2 CHRISTIAN AEBI AND GRANT CAIRNS A fascnatng account of the hstory of proofs of Wlson s and Fermat s theorems s gven n [, Chap. 3]. Although Theorem s a useful basc prmalty test, ts converse s false; for example, 34 (mod 34, but 34 s not prme; such numbers are called pseudoprmes. Some other pseudoprmes are: 56, 645, 05, 387, and 79, just to stop on a famous number. In a smlar ven, we have: Theorem. If p s an odd prme, then ( C (mod p. It seems surprsng that the above connecton doesn t seem to have been prevously explctly observed, especally snce Catalan numbers are the subject of such nterest (sometmes known as Catalan dsease and there have been so many proofs of Wlson s theorem, ncludng proofs by Catalan hmself [, Chap. 3]. We gve two proofs of Theorem. Proof of Theorem. Suppose that p s an odd prme. Modulo p, one has p, for all. Hence (p! (! and so C = p+ (( ( = p + (p!! (( ( p + (. Before gvng the next proof, frst recall the followng elementary facts (part (a was observed by Lebnz [, p. 59], part (b was observed as early as 830 [, p. 67] and appears for example n [, Ex ]: Lemma. If p s prme and 0 < < p, then we have: (a ( p 0 (mod p, (b ( ( (mod p, (c ( + ( p p (mod p, where denotes the multplcatve nverse of modulo p. Proof. Part (a s obvous, but that won t stop us gvng a proof n Secton 4. (b The bnomal theorem gves ( = ( p (. It follows that ( = ( p ( p + ( p + ( (, usng (a. Part (c follows from (b snce p ( p = (. Proof of Theorem. By Lemma (b, ( C = ( p + (. Notce that the above proofs are completely elementary; they don t even use Wlson s theorem. Lke Theorem, Theorem gves a necessary condton for p to be prme, and lke Theorem, ts converse s false; for example, C 953 (mod 5907, but 5907 s not prme (beng equal to We wll call such numbers Catalan pseudoprmes. Comparng Theorems and, notce that although the computaton of the Catalan

3 CATALAN NUMBERS, PRIMES AND TWIN PRIMES 3 numbers s qute nvolved [7], C s consderably smaller than p. Moreover, Catalan pseudoprmes seem to be far less common than standard pseudoprmes. Indeed, searchng for Catalan pseudoprmes wth a computer can be qute dscouragng. A more theoretcal approach conssts n tryng to dentfy Catalan pseudoprmes of a gven form, the smplest of all beng p, where p s prme. In that case the natural queston arses as to whether they would also be standard pseudoprmes. The affrmatve answer here below s even more precse. Proposton. If p s an odd prme, then the followng numbers are equal modulo p : (a C ( p, (b (, (c p, (d p, (e + p ( p, where denotes the nverse of modulo p. Proof. Frst note that f < p and s not a multple of p, then has an nverse modulo p, and p. Thus C p = p + ( p p ( p p = p ( p p p p p p p 3p p p 3p = ( p p p p = ( p p ( = ( p 3 3 ( p p+. p 3 3 p p p p + p So (a=(b. The clam (b=(e can be deduced drectly from [0, Theorem 33]. We supply a proof for completeness. We have ( ( So, expandng n powers of p, ( ( = ( p p p 3 3 = p p 3 p 3 p ( p p ( p p. (mod p. Modulo p, each nverse equals a unque number j wth j < p. Thus one has the followng fact observed by Cauchy [, Chap. III]: ( p = (p = pp 0 (mod p.

4 4 CHRISTIAN AEBI AND GRANT CAIRNS Hence ( ( + p ( p (mod p Thus (b=(e. The equaton (e=(c was apparently proved by Sylvester [8, Chap. 8A]; t follows from [0, Theorem 3], but once agan we supply a proof for completeness. Usng Lemma (c and (, we have modulo p, [( p ( p = ( + p p = j + p j= p ( + p (mod p. p ( ] p Ths gves (e=(c. Fnally, by Theorem, p (mod p, so modulo p, we have p = + xp, for some x {0,,..., p }. Thus Hence (c=(d. p = ( p p = ( + xp p p (mod p. Recall that f p s prme and p modulo p, then p s a Weferch prme; 093 and 35 are the only known Weferch prmes; there are no other Weferch prmes less than [], but at present t s not known whether there are only fntely many Weferch prmes. Weferch showed n 909 that f the Fermat equaton x p + y p = z p had a soluton for an odd prme p not dvdng xyz, then the smallest such p s necessarly a Weferch prme [7]. Notce that the above proposton has the followng corollary: Corollary. If p s prme, then the followng are equvalent: (a p s a Weferch prme, (b p s a pseudoprme, (c p s a Catalan pseudoprme. So = 093 and 37 = 35 are examples of Catalan pseudoprmes. Much more rare than standard pseudoprmes, 5907, 093 and 35 are the only Catalan pseudoprmes we are currently aware of. Remark. Notce that by Theorems and, when p s prme, ( C p (mod p. Proposton gves a stronger verson of ths. Indeed, from Proposton we obtan ( C = ( p + ( p + (p ( p( p (mod p. We remark n passng that an even stronger statement can be deduced from a result of Frank Morley [3]: f p > 3 s prme, then ( C For related nformaton, see [4, Lecture ]. ( p + p (mod p 3.

5 CATALAN NUMBERS, PRIMES AND TWIN PRIMES 5 Remark. As we saw above, Theorem s a trval consequence of Lemma. Although the converse of Theorem s false, the converse of Lemma s true, as was observed by Lebnz [, p. 9]: a natural number p s prme f and only f ( p s dvsble by p for all 0 < < p. Indeed, f n s composte, and q s a prme dvsor of n, then ( n q sn t dvsble by n, as one can see by wrtng n ( n q = (n (n... (n q + q! and notng that snce q dvdes n, q doesn t dvde any of the terms n the numerator. 3. TWIN PRIMES. There s a twn prme verson of Wlson s theorem, known as Clement s theorem [8], that says: Clement s Theorem. The natural numbers p, p + are both prme f and only f 4.(p! + p (mod p(p +. Clement s theorem has been redscovered and generalzed by a number of people [5, 4]. In fact, t was dscovered by Zahlen a few years before Clement [30]. An alternate expresson s gven n [9]. There s an obvous Fermat verson of Clement s theorem, whch we haven t notced n the lterature: Theorem 3. If natural numbers p, p+ are both prme, then p+ 3p+8 (mod p(p+. Proof. Suppose that p, p + are both prme. We are requred to show that p+ 3p + 8 (mod p and p+ 3p + 8 (mod p +. Modulo p, Theorem gves whle modulo p +, Theorem gves as requred. p+ 8 3p + 8 (mod p, p+ 3(p + + = 3p + 8 (mod p +, The converse to Theorem 3 s false. For example, for p = 56, one has p+ 3p + 8 (mod p(p+, but whle 56 s prme, 563 s a pseudoprme. Another example s p = 4369, where p and p + are both pseudoprmes. Nevertheless, n the same way that Fermat s lttle theorem has a generalzaton of the form : p s prme f and only f for every prme q < p, q (mod p, Theorem 3 can also be generalzed to: Theorem 4. Natural numbers p and p + are both prme, f and only f for all prmes q < p, q p+ p(q + q (mod p(p +. Theorem 4 can be establshed n the same way we proved Theorem 3, wth the help of lttle Fermat s extenson. Returnng to Catalan numbers, there s a recent twn-prme crtera that s not entrely unrelated to the Catalan numbers []. In a dfferent drecton, the followng observaton s drectly analogous to Clement s theorem and Theorem 3: Theorem 5. If natural numbers p, p + are both prme, then 8( C 7p + 6 (mod p(p +.

6 6 CHRISTIAN AEBI AND GRANT CAIRNS Proof. Suppose that p, p + are both prme. Modulo p, Theorem gves One has C 8( C 8( C = p+3 C 4p p+ whch completes the proof. 8 7p So, modulo p +, Theorem gves = p + 3 p+ ( C p+ p 4 p + 3 p (mod p. 7p + 6 (mod p +, We don t have a counter-example to the converse of Theorem 5; the only Catalan pseudoprmes we currently know are 5907, and 37, and none of the numbers 5907 ±, ±, 37 ± s prme. Remark 3. Usng Lemma and Remark, t s not dffcult to establsh the followng: the natural numbers p, p + are both prme f and only f ( p ( + + p (mod p(p +, for all 0 < < p. 4. A DOOR AJAR?. The connecton between prmes and Catalan numbers opens the door (however narrowly to possble connectons between prmes and varous combnatoral problems. There are precedents for ths sort of thng. There s a geometrc proof of Fermat s lttle theorem; see [3, Ch. 3.]. Here s one way of seeng t. Consder the possble black-whte colourngs of the vertces of a regular p-gon. There are p such colourngs. The cyclc group Z p acts by rotaton on the polygon and hence on the set of colourngs. There are two colourngs fxed by ths acton (all black and all whte, and for p prme, the Z p -acton s free on the set of the p other colourngs. Thus p 0 (mod p, whch proves Theorem. There are other elementary results that can be establshed n a smlar manner. For example, there are ( p colourngs of the above knd n whch exactly vertces are coloured black. For p prme and 0 < < p, the Z p -acton on these colourngs s obvously free, so ( p 0 (mod p, whch proves Lemma (a. Lemma (a s admttedly trval, beng mmedate from the defnton ( p = p!. (p!! However, the same dea can be used to gve a more nterestng result. Consder the regular mp-gon, where m N. The vertces can be grouped nto p lots of m consecutve vertces, whch one can regard as formng the sdes of a p-gon. Now consder the possble blackwhte colourngs of vertces of the mp-gon. The acton of Z p s once agan free outsde the fxed ponts. The colourngs that are fxed by the acton are just those colourngs that are dentcal on each edge of the p-gon. So one has mmedately: Proposton. If p s prme, then for all m N (a pm 0 (mod p f 0 (mod p, (b ( pm p ( m (mod p for all N. From Proposton we can quckly deduce Lucas Theorem [, Secton XXI]: Lucas Theorem. If p s prme and 0 n, j < p, then ( pm+n p+j ( m ( n j (mod p.

7 CATALAN NUMBERS, PRIMES AND TWIN PRIMES 7 Proof. By Proposton (a we can assume that n > 0. Then for the case j = 0, ( pm+n pm + n ( ( p = pm+n p pm+n p (mod p, p(m + n and the result follows by nducton on n. For j 0, ( pm+n pm + n ( p+j = pm+n n p+j p + j j ( pm+n p+j and agan the requred s obtaned by nducton on n. (mod p, 5. BACK TO THE MIDDLE BINOMIAL COEFFICIENT. For convenence, let us ntroduce the followng notaton: ( γ n := ( n n n, for odd n. Theorem can be rephrased as follows: f p s an odd prme, then γ p (mod p. The equaton (a=(b of Proposton can be rewrtten as follows: f p s an odd prme, then γ p γ p (mod p. One also has: Lemma. (a If p s an odd prme, then γ mp γ m (mod p for all odd m N. (b If p, q are dstnct odd prmes, then γ pq γ p γ q γ p + γ q (mod pq. (c If p s an odd prme, then for all odd n p, γ n 0 and γ n (n γ p n+ (mod p. Proof. (a Argung as n the proof of Lemma (b, one has for all ( ( m = ( mp + ( mp + ( ( mp and so by Proposton, ( ( m ( m + ( m + ( j ( m j, (mod p where j = p. For = m one has p = m p = m. So ( ( m m ( m + ( m + ( m ( m m = ( m ( m m. That s, γ mp γ m (mod p. (b Suppose that p, q are dstnct odd prmes. From part (a, γ pq γ q (mod p. So, as γ p (mod p, we have γ pq γ p γ q (mod p. Smlarly, γ pq γ p γ q (mod q. Thus γ pq γ p γ q (mod pq. Furthermore, snce γ p 0 (mod p and γ q 0 (mod q, one has (γ p (γ q 0 (mod pq, and so γ p γ q γ p + γ q (mod pq, as requred. (c Frst notce that for all odd > Hence ( γ = ( = 4 ( 3 3 ( γ = 4 γ. 3 = 4 ( γ.

8 8 CHRISTIAN AEBI AND GRANT CAIRNS Now we prove (c by nducton on n. For n = t s obvous, so let n >. Usng ( frst and the nducton hypotheses we obtan γ n 4 n n (n 3 γ p n+3 (mod p = 4 n n (n 3 4 p n + p n + γ p n+ = n n p n + p n + (n γ p n+ (n γ p n+ (mod p (usng ( agan Remark 4. It s not true that γ pqr γ p γ q γ r (mod pqr for all dstnct prmes p, q, r. For example, γ 05 γ 3 γ 5 γ 7 (mod 05. Notce that from the defnton, a composte number n s a Catalan pseudoprme f and only f γ n (mod n. Further, one has: Proposton 3. If p, q are dstnct odd prmes, then pq s a Catalan pseudoprme f and only f γ q (mod p and γ p (mod q. Proof. If pq s a Catalan pseudoprme, then γ pq (mod pq. In partcular, γ pq (mod p. So by Lemma (a, γ p γ q (mod p. But as p s prme, γ p (mod p. Hence γ q (mod p. By the same reasonng, γ p (mod q. Conversely, f γ p (mod q and γ p (mod q, then as p, q are dstnct prmes, γ p (mod pq. Smlarly, γ q (mod pq and so by Lemma (b, γ pq (mod pq. The above consderatons enable one to show that f p, q are prme wth p < q and ether p or q p s qute small, then pq s not a Catalan pseudoprme. To gve a trval example of ths, we prove: Corollary. There are no Catalan pseudoprmes of the form p(p +, where p, p + are twn prmes. Proof. If p, p + are prmes, then by Lemma (c, γ p ( γ 3 (mod p +. One has γ 3 = and by Fermat s lttle theorem, ( 4 (mod p +. So γ p 3 (mod p +. If p(p + was a Catalan pseudoprme, then by Proposton 3 we would have γ p (mod p + and thus 3 (mod p +,.e., p + = 9 contradctng the assumpton that p + s prme. We now come to the man result of ths paper, whch enables one to reduce the calculaton of γ n (mod p to the case where n < p. Theorem 6. If p s an odd prme, then for all odd n N, 0 ; n/p odd and n not a multple of p, γ n γ n/p ; n/p odd and n a multple of p, γ n/p +.γ n p n/p ; n/p even. (mod p

9 CATALAN NUMBERS, PRIMES AND TWIN PRIMES 9 Proof. If n/p s odd and n a multple of p, then n has the form mp where m s odd, and Lemma (a gves the requred result. If n/p s odd and n s not a multple of p, then n has the form mp + where m s odd and 0 < < p/. By nducton, mp + ( γ mp+ = 4 mp + ( + γ mp+( 0 and for =, Equaton ( gves: γ mp+ = 4 mp mp + γ mp 0 (mod p. (mod p. If n/p s even, then n has the form mp where m s odd and 0 < < p/. Applyng Equaton ( tmes gves: γ mp = ( 4 mp mp mp 4 mp 3... mp mp + γ mp ( γ mp Hence, snce γ mp γ m (mod p by Lemma (a, we have: (3 γ mp ( Notce that by defnton γ m γ + = ( (! ( = ( (! = ( (mod p. (mod p So, from Equaton 3, γ mp 4 γ + γ m (mod p. Thus, by Lemma (c, γ mp γ p γ m (mod p. That s, γ n γ n/p + γ n p n/p (mod p, as requred. Remark 5. Theorem 6 can be alternatvely deduced from Lucas Theorem. We menton also that the blocks of zeros where q/p s odd and q s not a multple of p, has also been observed for the Catalan numbers []. Usng Theorem 6 and Proposton 3, our calculatons show that there are no Catalan pseudoprmes less than 0 0 of the form pq, where p, q are dstnct prmes. Notce that Theorem 6(a gves a consderable extenson of Corollary. Indeed, f p, q are prme and p < q < p, then by Theorem 6, γ q 0 (mod p and so pq s not a Catalan pseudoprme, by Proposton 3. The frst possble case would therefore seem to be the stuaton where q = p +, but n fact there are no Catalan pseudoprmes of ths form ether: Corollary 3. There are no Catalan pseudoprmes of the form pq, where p, q = p + s a Sophe German par. Proof. Otherwse for q = p +, Theorem 6 gves γ q γ 3 γ (mod p. But Proposton 3 mples γ q (mod p. Hence p = 3 and q = 7. Agan by Proposton 3 we would have γ p (mod q, but γ 3 = (mod 7.

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11 CATALAN NUMBERS, PRIMES AND TWIN PRIMES 8., My numbers, my frends, Sprnger-Verlag, New York, Rchard P. Stanley, Enumeratve combnatorcs. Vol., Cambrdge Studes n Advanced Mathematcs, vol. 6, Cambrdge Unversty Press, Cambrdge, Jean Perre Zahlen, Sur un genre nouveau de crtères de prmalté, Eucldes, Madrd 6 (946, no. 64, Collège Calvn, Geneva, Swtzerland E-mal address: chrstan.aeb@edu.ge.ch Department of Mathematcs, La Trobe Unversty, Melbourne, Australa 3086 E-mal address: G.Carns@latrobe.edu.au