On the largest prime factor of numerators of Bernoulli numbers

ON THE LARGEST PRIME FACTOR OF NUMERATORS OF BERNOULLI NUMBERS ATTILA B´ ERCZES AND FLORIAN LUCA Abs ac . We p o e ha o mos n, he nume a o o he Be noulli numbe B2nis di isible by a la ge p ime. 2000 Ma hema ics Subjec Classi ica ion: P ima y 11B68 1. In oduc ion Fo a posi i e in ege n, we w i e ω(n) o he numbe o dis inc p ime ac o s o n. Le {Bn}n≥0be he sequence o Be noulli numbe s gi en by B0= 1 and Bn= 1 − n−1 X k=0 n kBk n−k+ 1 o all n≥1. Then B1=−1/2 and B2n+1 = 0 o all n≥0. Fu he mo e, we ha e (−1)n+1B2n>0. W i e B2n=: (−1)n+1Cn/Dnwi h cop ime posi i e in ege s Cnand Dn. The denomina o Dnis well-unde s ood by he on S aud –Clausen heo em which asse s ha Dn=Y p−1|2n p. (1) As o Cn, i was p o ed in [3] ha he es ima e ω Y n≤x Cn!≥(1 + o(1)) log x log log xholds as x→ ∞. He e, we look a he la ges p ime ac o o Cn. Fo a posi i e in ege mwe pu P(m) o he la ges p ime ac o o m. The esea ch was suppo ed in pa by p ojec SEP-CONACyT 79685, by g an s T67580 and T75566 o he Hunga ian Na ional Founda ion o Scien i ic Resea ch. The wo k is suppo ed by he T´ AMOP 4.2.1./B-09/1/KONV-2010-0007 p ojec . The p ojec is implemen ed h ough he New Hunga y De elopmen Plan, co- inanced by he Eu opean Social Fund and he Eu opean Regional De elopmen Fund. F. L. wo ked on his p ojec while he isi ed he Ins i u e o Ma hema ics o he Uni e si y o Deb ecen, Hunga y in Augus 2011. He hanks he membe s o ha depa men o hei hospi ali y. 1 2 A. B´ ERCZES AND F. LUCA Theo em 1. The inequali y P(Cn)>1 4log n holds o mos posi i e in ege s n. He e and in wha ollows, we use he symbols Oand owi h hei usual meaning. We also use c1, c2, . . . o compu able posi i e cons an s and x0 o a la ge eal numbe , no necessa ily he same om one occu ence o he nex . P oo . We le xbe la ge. Pu M(x) := {x/2≤n≤x:P(Cn)≤(1/4) log x}.(2) Pu y:= xlog log log x/ log log x. We le L1(x) := {n≤x:P(n)≤y}.(3) I is known (see Chap e III.5 in [5]), ha #L1(x) = xexp(−(1 + o(1))ulog u),whe e u:= log x log y. Since o us u= log log x/ log log log x, we ge easily ha #L1(x) = Ox (log x)1/2.(4) We le τ(m) s and o he numbe o di iso s o m. We pu L2(x) := {n≤x:τ(n)>(log x)2}.(5) Since X n≤x τ(n) = O(xlog x), (see Theo em 320 on Page 347 in [2]), i ollows easily ha #L2(x) = Ox log x.(6) Le L3(x) := {n≥x:p−1|2n o some p ime pwi h P(p−1) > y}.(7) The p oo o Theo em 1.1 in [1] shows ha #L3(x) = Ox (log x)0.05 .(8) F om now on, we look a in ege s nin N(x) := M(x) ∪3 i=1 Li(x).(9) ON THE LARGEST PRIME FACTOR OF NUMERATORS OF BERNOULLI NUMBERS3 Pu z:= (log x)2and le Ibe an a bi a y in e al in [x/2, x] o leng h a mos z. Pu T:= (1/4) log xand pu K:= π(T). We show ha o x > x0,Icon ains less han K+ 3 numbe s om N(x). Assume i s ha we ha e p o ed his and le us see how o inish he a gumen . Then #N(x)≤x−x/2 (log x)2+ 1(K+ 2) = Ox (log x)2·T log T =Ox log xlog log x,(10) which oge he wi h es ima es (4), (6), (8) shows ha #M(x)≤#L1(x)+#L2(x)+#L3(x)+#N(x) = Ox (log x)0.05 . (11) The desi ed es ima e now ollows by eplacing xwi h x/2, hen wi h x/4, e c., and summing up he esul ing es ima es (11). I emains o p o e ha indeed Icanno con ain K+ 3 numbe s om N(x) o x > x0. Assume ha i does and le hem be n1< n2<· · · < nK+3. Pu λi:= ni−n1 o i= 1, . . . , K + 3. Then 0 = λ1< λ2<· · · < λK+3 ≤z. Le n=ni o some i= 1, . . . , K + 3. We use he o mula ζ(2n) = (−1)n+1B2n (2π)2n 2(2n)! =Cn(2π)2n Dn2(2n)!, as well as he ap oxima ion ζ(2n) = 1 + 1 22n+1 32n+· · · = 1 + O1 22n, o ge ha Cn=Dn 2(2n)! (2π)2nζ(2n) = Dn 2(2n)! (2π)2n1 + O1 22n.(12) We ake loga i hms in (12) abo e o a i e a log Cn−log Dn−log(2(2n)!)+2nlog(2π) = log 1 + O1 22n=O1 2x. (13) We now le pj o j= 1, . . . , K be all he p imes p≤Tand w i e Cni=pαi,1 1pαi,2 2· · · pαi,K K o all i= 1, . . . , K + 3. 4 A. B´ ERCZES AND F. LUCA Obse e ha since τ(2n)≤2τ(n)≤2(log x)2, we ha e ha Dn=Y p−1|2n p≤(2n+1)τ(2n)≤(2x+1)2(log x)2<exp(3(log x)3) (x > x0). (14) Thus, om o mula (12), we ha e ha Cn≤Dn 2(2n)! (2π)2nζ(2) ≤2ζ(2)Dn (2π)2n(2n)2n<2ζ(2)Dn π2nn2n <2ζ(2) exp(3(log x)3) πxx2x< x2x o x>x0, which implies ha αi,j ≤2xlog x log pj ≤2xlog x log 2 <3xlog x o all 1 ≤i≤K+3,1≤j≤K. Le ∆:= (∆1,...,∆K+3) be a nonze o ec o in he null-space o he (K+ 2) ×(K+ 3) ma ix A=         a1,1a2,1· · · aK+3,1 a1,2a2,2· · · aK+3,2 . . .. . .· · · . . . a1,K a2,K · · · aK+3,K 1 1 · · · 1 n1n2· · · nK+3         . Such a ec o exis s and can be compu ed wi h C ame ’s ule. I ’s heigh sa is ies max{|∆i|}1≤i≤K+3 ≤(K+ 2)! max{|αi,j|,|n`|, i, j, `}K+2 <(3x(K+ 2) log x))K+2 <(3x(log x)2)π(T)+2 < x2(π(T)+2) <exp((log x)2),(15) o x>x0. We now e alua e o mula (13) in n=ni o i= 1, . . . , K +3 and ake he linea combina ion wi h coe icien s ∆1,...,∆K+3 o he esul ing ela ions ge ing  K+3 X i=1 ∆ilog Cni− K+3 X i=1 ∆ilog Dni− K+3 X i=1 ∆ilog(2(2ni)! + K+3 X i=1 2∆inilog(2π) =O PK+3 i=1 |∆i| 2x!.(16) ON THE LARGEST PRIME FACTOR OF NUMERATORS OF BERNOULLI NUMBERS5 In he le –hand side o es ima e (16) abo e, he i s sum anishes; i.e., K+3 X i=1 ∆ilog Cni= 0, because he ec o ∆is o hogonal o he i s K ows o A. Simila ly, he las sum also anishes; i.e., K+3 X i=1 ∆ini= 0, because ∆is o hogonal o he las ow o A. Finally, w i ing 2(2ni)! = 2(2n1)!(2n1+1)(2n1+2) · · · (2ni) =: 2(2n1)!Xi(i= 1, . . . , K+3), we ge ha log(2(2ni)!) = log(2(2n1)!) + log Xi. Hence, K+3 X i=1 ∆ilog(2(2ni)!) = K+3 X i=1 ∆ilog(2(2n1)!)+ K+3 X i=1 ∆ilog Xi= K+3 X i=1 ∆ilog Xi, (17) whe e we used PK+3 i=1 ∆i= 0, because ∆is o hogonal o he i s be o e las ow o ma ix A. Thus using also (15), es ima e (16) becomes  K+3 X i=1 ∆ilog(Dni/Xi) =O(K+ 3) exp((log x)2) 2x=O1 2x/2. (18) In he le –hand side o es ima e (18) we ha e a linea o m in loga- i hms. Fu he , Xi<(2x)2(ni−n1)≤(2x)2z<exp(3(log x)3) (x>x0),(19) which is he same es ima e as es ima e (14) wi h Dni eplaced by Xi o all i= 1, . . . , K + 3. Fo each i= 1, . . . , K + 3, le Pi:= P(ni). Then Pi|Xi. Also, Pidoes no di ide Dnj o any j= 1, . . . , K + 3. Indeed, o he wise he e would exis q:= Pisuch ha o some j, we ha e ha q|Dnj. Thus, he e exis s a p ime numbe psuch ha q|p−1 and p−1|2nj. Howe e , his is no possible because nj6∈ L3(x). Also, Pi di ides Xj o all j≥ibu does no di ide Xj o any j < i. Indeed, his las claim ollows because i Pi|Xj o some j < i, hen he e exis s m∈[2n1,2nj] such ha Pi|m. Bu also Pi|ni, so Pi|2ni−m, and his las numbe is nonze o since 2ni6∈ [2n1,2nj]. Howe e , his is no possible o la ge xsince i would lead o y < Pi≤2ni−m≤2z, which is impossible o x>x0. This shows ha he linea o m appea ing in 6 A. B´ ERCZES AND F. LUCA he le –hand side o (17) is nonze o (indeed, i iis maximal such ha ∆i6= 0, hen he coe icien o log Piin he le is exac ly ∆i6= 0). We apply a linea o m in loga i hms ´a la Bake in he le –hand side o (18) (see [4], o example). We ge ha he le –hand side o (18) is a leas >exp −c1cK 2 K+3 Y i=1 max{log Dnilog Xni}!log max{|∆i|}!, o some app op ia e cons an s c1and c2. Wi h he bounds (14), (19) and (15), he abo e exp ession is a leas >exp −c1cK 2(3(log x)3)K+3(log x)2, which compa ed wi h (18) gi es x(log 2)/2−c3< c1(3c2(log x)3)K+3(log x)2, wi h some app op ia e cons an c3. This las es ima e implies easily ha he inequali y K > (1/3−ε) log x/ log log xholds o all ε > 0 and x > x0(depending on ε). Taking a su icien ly small alue o ε (say ε:= 1/100), and in oking he P ime Numbe Theo em o es ima e K=π(T), we ge a con adic ion. This inishes he a gumen and he p oo o he heo em.  Re e ences [1] J. F iedlande and F. Luca, “On he alue se o he Ca michael λ- unc ion”, J. Aus al. Ma h. Soc. 82 (2007), 123–131. [2] G. H. Ha dy and E. M. W igh , An in oduc ion o he heo y o numbe s, Ox o d Uni e si y P ess, Ox o d, six h edi ion, 2008. Re ised by D. R. Hea h- B own and J. H. Sil e man. [3] F. Luca and A. Piza o, “Some ema ks on he alues o he Riemann ze a unc ion and Be noulli numbe s”, P ep in , 2011. [4] E. M. Ma ee , “An explici lowe bound o a homogeneous a ional linea o m in loga i hms o algeb aic numbe s. II”, Iz . Ross. Akad. Nauk Se . Ma . 64 (2000), 125–180; English ansl. in Iz . Ma h. 64 (2000), 1217–1269. [5] G. Tenenbaum, In oduc ion o analy ic and p obabilis ic numbe heo y, Uni- e si y P ess, Camb idge, UK, 1985. ON THE LARGEST PRIME FACTOR OF NUMERATORS OF BERNOULLI NUMBERS7 A. B´ e czes Ins i u e o Ma hema ics, Uni e si y o Deb ecen Numbe Theo y Resea ch G oup, Hunga ian Academy o Sciences and Uni e si y o Deb ecen H-4010 Deb ecen, P.O. Box 12, Hunga y E-mail add ess:[email p o ec ed] F. Luca Ins i u o de Ma em´ a icas Uni e sidad Nacional Au onoma de M´ exico C.P. 58089, Mo elia, Michoac´ an, M´ exico E-mail add ess:[email p o ec ed]