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 Abstract. We prove that for most n, the numerator of the Bernoulli number B2nis divisible by a large prime. 2000 Mathematics Subject Classification: Primary 11B68 1. Introduction For a positive integer n, we write ω(n) for the number of distinct prime factors of n. Let {Bn}n≥0be the sequence of Bernoulli numbers given by B0= 1 and Bn= 1 − n−1 X k=0 n kBk n−k+ 1 for all n≥1. Then B1=−1/2 and B2n+1 = 0 for all n≥0. Furthermore, we have (−1)n+1B2n>0. Write B2n=: (−1)n+1Cn/Dnwith coprime positive integers Cnand Dn. The denominator Dnis well-understood by the von Staudt–Clausen theorem which asserts that Dn=Y p−1|2n p. (1) As for Cn, it was proved in [3] that the estimate ω Y n≤x Cn!≥(1 + o(1)) log x log log xholds as x→ ∞. Here, we look at the largest prime factor of Cn. For a positive integer mwe put P(m) for the largest prime factor of m. The research was supported in part by project SEP-CONACyT 79685, by grants T67580 and T75566 of the Hungarian National Foundation for Scientific Research. The work is supported by the T´ AMOP 4.2.1./B-09/1/KONV-2010-0007 project. The project is implemented through the New Hungary Development Plan, cofinanced by the European Social Fund and the European Regional Development Fund. F. L. worked on this project while he visited the Institute of Mathematics of the University of Debrecen, Hungary in August 2011. He thanks the members of that department for their hospitality. 1 2 A. B´ ERCZES AND F. LUCA Theorem 1. The inequality P(Cn)>1 4log n holds for most positive integers n. Here and in what follows, we use the symbols Oand owith their usual meaning. We also use c1, c2, . . . for computable positive constants and x0for a large real number, not necessarily the same from one occurrence to the next. Proof. We let xbe large. Put M(x) := {x/2≤n≤x:P(Cn)≤(1/4) log x}.(2) Put y:= xlog log log x/ log log x. We let L1(x) := {n≤x:P(n)≤y}.(3) It is known (see Chapter III.5 in [5]), that #L1(x) = xexp(−(1 + o(1))ulog u),where u:= log x log y. Since for us u= log log x/ log log log x, we get easily that #L1(x) = Ox (log x)1/2.(4) We let τ(m) stand for the number of divisors of m. We put L2(x) := {n≤x:τ(n)>(log x)2}.(5) Since X n≤x τ(n) = O(xlog x), (see Theorem 320 on Page 347 in [2]), it follows easily that #L2(x) = Ox log x.(6) Let L3(x) := {n≥x:p−1|2nfor some prime pwith P(p−1) > y}.(7) The proof of Theorem 1.1 in [1] shows that #L3(x) = Ox (log x)0.05 .(8) From now on, we look at integers nin N(x) := M(x)\ ∪3 i=1 Li(x).(9) ON THE LARGEST PRIME FACTOR OF NUMERATORS OF BERNOULLI NUMBERS3 Put z:= (log x)2and let Ibe an arbitrary interval in [x/2, x] of length at most z. Put T:= (1/4) log xand put K:= π(T). We show that for x > x0,Icontains less than K+ 3 numbers from N(x). Assume first that we have proved this and let us see how to finish the argument. 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 together with estimates (4), (6), (8) shows that #M(x)≤#L1(x)+#L2(x)+#L3(x)+#N(x) = Ox (log x)0.05 . (11) The desired estimate now follows by replacing xwith x/2, then with x/4, etc., and summing up the resulting estimates (11). It remains to prove that indeed Icannot contain K+ 3 numbers from N(x) for x > x0. Assume that it does and let them be n1< n2<· · · < nK+3. Put λi:= ni−n1for i= 1, . . . , K + 3. Then 0 = λ1< λ2<· · · < λK+3 ≤z. Let n=nifor some i= 1, . . . , K + 3. We use the formula ζ(2n) = (−1)n+1B2n (2π)2n 2(2n)! =Cn(2π)2n Dn2(2n)!, as well as the aproximation ζ(2n) = 1 + 1 22n+1 32n+· · · = 1 + O1 22n, to get that Cn=Dn 2(2n)! (2π)2nζ(2n) = Dn 2(2n)! (2π)2n1 + O1 22n.(12) We take logarithms in (12) above to arrive at log Cn−log Dn−log(2(2n)!)+2nlog(2π) = log 1 + O1 22n=O1 2x. (13) We now let pjfor j= 1, . . . , K be all the primes p≤Tand write Cni=pαi,1 1pαi,2 2· · · pαi,K Kfor all i= 1, . . . , K + 3. 4 A. B´ ERCZES AND F. LUCA Observe that since τ(2n)≤2τ(n)≤2(log x)2, we have that 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, from formula (12), we have that 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< x2xfor x>x0, which implies that αi,j ≤2xlog x log pj ≤2xlog x log 2 <3xlog xfor all 1 ≤i≤K+3,1≤j≤K. Let ∆:= (∆1,...,∆K+3) be a nonzero vector in the null-space of the (K+ 2) ×(K+ 3) matrix 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 vector exists and can be computed with Cramer’s rule. It’s height satisfies 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) for x>x0. We now evaluate formula (13) in n=nifor i= 1, . . . , K +3 and take the linear combination with coefficients ∆1,...,∆K+3 of the resulting relations getting  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 the left–hand side of estimate (16) above, the first sum vanishes; i.e., K+3 X i=1 ∆ilog Cni= 0, because the vector ∆is orthogonal to the first Krows of A. Similarly, the last sum also vanishes; i.e., K+3 X i=1 ∆ini= 0, because ∆is orthogonal to the last row of A. Finally, writing 2(2ni)! = 2(2n1)!(2n1+1)(2n1+2) · · · (2ni) =: 2(2n1)!Xi(i= 1, . . . , K+3), we get that 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) where we used PK+3 i=1 ∆i= 0, because ∆is orthogonal to the first before last row of matrix A. Thus using also (15), estimate (16) becomes  K+3 X i=1 ∆ilog(Dni/Xi) =O(K+ 3) exp((log x)2) 2x=O1 2x/2. (18) In the left–hand side of estimate (18) we have a linear form in logarithms. Further, Xi<(2x)2(ni−n1)≤(2x)2z<exp(3(log x)3) (x>x0),(19) which is the same estimate as estimate (14) with Dnireplaced by Xifor all i= 1, . . . , K + 3. For each i= 1, . . . , K + 3, let Pi:= P(ni). Then Pi|Xi. Also, Pidoes not divide Dnjfor any j= 1, . . . , K + 3. Indeed, otherwise there would exist q:= Pisuch that for some j, we have that q|Dnj. Thus, there exists a prime number psuch that q|p−1 and p−1|2nj. However, this is not possible because nj6∈ L3(x). Also, Pi divides Xjfor all j≥ibut does not divide Xjfor any j < i. Indeed, this last claim follows because if Pi|Xjfor some j < i, then there exists m∈[2n1,2nj] such that Pi|m. But also Pi|ni, so Pi|2ni−m, and this last number is nonzero since 2ni6∈ [2n1,2nj]. However, this is not possible for large xsince it would lead to y < Pi≤2ni−m≤2z, which is impossible for x>x0. This shows that the linear form appearing in 6 A. B´ ERCZES AND F. LUCA the left–hand side of (17) is nonzero (indeed, if iis maximal such that ∆i6= 0, then the coefficient of log Piin the left is exactly ∆i6= 0). We apply a linear form in logarithms ´a la Baker in the left–hand side of (18) (see [4], for example). We get that the left–hand side of (18) is at least >exp −c1cK 2 K+3 Y i=1 max{log Dnilog Xni}!log max{|∆i|}!, for some appropriate constants c1and c2. With the bounds (14), (19) and (15), the above expression is at least >exp −c1cK 2(3(log x)3)K+3(log x)2, which compared with (18) gives x(log 2)/2−c3< c1(3c2(log x)3)K+3(log x)2, with some appropriate constant c3. This last estimate implies easily that the inequality K > (1/3−ε) log x/ log log xholds for all ε > 0 and x > x0(depending on ε). Taking a sufficiently small value for ε (say ε:= 1/100), and invoking the Prime Number Theorem to estimate K=π(T), we get a contradiction. This finishes the argument and the proof of the theorem.  References [1] J. Friedlander and F. Luca, “On the value set of the Carmichael λ-function”, J. Austral. Math. Soc. 82 (2007), 123–131. [2] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford University Press, Oxford, sixth edition, 2008. Revised by D. R. HeathBrown and J. H. Silverman. [3] F. Luca and A. Pizarro, “Some remarks on the values of the Riemann zeta function and Bernoulli numbers”, Preprint, 2011. [4] E. M. Matveev, “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II”, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125–180; English transl. in Izv. Math. 64 (2000), 1217–1269. [5] G. Tenenbaum, Introduction to analytic and probabilistic number theory, University Press, Cambridge, UK, 1985. ON THE LARGEST PRIME FACTOR OF NUMERATORS OF BERNOULLI NUMBERS7 A. B´ erczes Institute of Mathematics, University of Debrecen Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen H-4010 Debrecen, P.O. Box 12, Hungary E-mail address:[email protected] F. Luca Instituto de Matem´ aticas Universidad Nacional Autonoma de M´ exico C.P. 58089, Morelia, Michoac´ an, M´ exico E-mail address:[email protected]