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]