108
Onamicbleurs
.
ToprfesLázlóKamronhis50tbday
.
By P . ERDÖS in Jerusalem .
Denotbyr()hsumofaldivrsn(clude)
.Twonumbers
andbreclami f()-ab=+
numbers20ad84wherladyknowtPYTHAGORS
thenamicbluersthaif
.Thesmalticb
. The origin of
c1,(a)_d
dia
a.
thenadbrmiclefandoyi,()=ba1knowiftheranilymacbenurs
this rue,nfacti bojeurdtha nmberoficaluberslthanigre thanl-Eforevya>0ifno(F)
KANOLD1)gavenuprstimaefohnubramicle s
factheprovd tensiyofamcbluersi than0
otherwdspovthaednsiyoftgera whicterxs
abstifyng()-ab+islethan024
. It is not yet
. It seems likely that
.Recntly
. In
.204, in
.
Intheprs owhalprvet hdnsiyofteamcbl
numbersi0
numberslthanies c
.Byourmethdwclprovetha numbroficale
•n/log log log n ;verylikmuchoes
n
(log n)' ,
true,hnmbofaicleurs thaniodub
forevyk,but mehodsntemuiablfortngsucha
good result .
AnoldcjeturOfCATLNstaehifwput
a1 (a)-a,(air-1~(a))
thenforvyathesqunc,"'(a)isbounde
a(") 1forsmen,thquceai'?()sprodicaftemng,i
necsary,fitnumbero sithebgn
. In other words either
.Itwouldnbehar
1) H . J .KANOLD,ÜberdiasymptocheDivngwseZahlmng
cedingsofth erainlcogsfmateicnsArdam1954,p
. Pro. 30.
P . Erdös : Onamicbleurs
.
109
toprvebyumthodafrevykthdnsioftegr whic
aik' (a}
= a is 0.2) Ontheoradqustionlkehabvmntioed
CATALAN, orwhet dnsiyoftegra, whicter
conjeturOf
is an n with a ;') (a)=1, exist m'nacibletprsn
We shall need several Lemmas.
.
Lemma 1 . Let q i beasquncofprimesatyng
1 = a . Denote
i=1 qi
v,,(n) thenumbrofq'swhicdven
< A is 0 for every A .
by
. Then the density of integers n with
V, (n)
Lema1ispcleofathrm
TURÁN3)
a weaker form that if 0 - ip(p) < K for all primes p,
ip(n)= Y v(p)
i
p ,,.
factors
n,
thenforalbu
whicasertn
?P(P) =
S
p p
x
and
where the summation is extended over all different prime
o(N) values - N of n we have
:s
4
(1)
P)
n
r=~
P
<
=~
p
1
Intheprscawhveonlytakip()_=Ifbelongstur
qi
sequnc,adp()=0forthe pims
p.
Lemma 2 . LetAbanycos
. Then the density of integers n for
whic
a(n)0
(mod (Hp)-21 )
p-A
is 0 for every A .
Itsuficeohwtaednsiyoftger
a p - A
q 1 i q_,, . . .
forwhic
n forwhictexs
(mod p-1) is 0. Let B beanrityeg,
the primes satisfying q - - I (mod p), qi > B . It is well known that
a(n)0
1 - a . ThusbyLemaItdnsiyofhetgrsdivbleyfwrthan
A
i=1 q ,
of the q's is 0.Alsothedniyf tegrsdivblythesquarofn
1,
the q's is clearytmos
-1 q,
the q's andbyo
chosenarbitlyge,hdnsityofe grswith
thenclary
q 2,
2) Itcanbeojurd
Q ;'(a)= a and u, t '(a) =
Historyfhe onumbrs1,p
for a = 14316, k = 28 .(Ibid
3) P . TURÁN, ÜberingValem rung
nujan . J. London Math .Soc
a
< B.
Now if n is
a(n)=-=0
divsbleyat Aof
(mod p-;) . Thusince
canbe
a(n)-0 (mod p- 1) is 0 .
B
that for every k there exists infinitely many a's with
-.
1<k This aproblemf0
. MEISSNER . (See : DICKSON,
. 49.) POULET observdthaf
a- 12496, k = 5 and
. p . 50.)
eines Satzes von Hardy and Rama. 11 (1936), 125-133.
for
I
P. Erdös
110
Lemma 3 . Denote
-'n
aA(n)= d~
.
d-A
Then to every e and ri therxisanA,othfr>A,enumbrof
> rn
integrs<xfowhica(n)-A
is less than i x.
We evidently have
(2)
,i--1
(a(n} - aA(n) )
=
n
Z
?i-=1,
,,
d-A
d
= d1 >A
d., <
d,-X,:d t
IfLema3wouldntberwouldhave
,Fx
1_- .A
a(n)-aA(n)
integrsdoxcing
x' x'
< -.
d`
A
rn for at least
. Thus
(3)
-1
Thus for A >
(a(n)-aA(n))
4 :, (3) contradis
1 8-
> r d~F
d >
'iE4x2 .
(2) proving so Lemma 3 .
Theorem . Thednsityofamcbleursi0
Denotby(a
.
i , b i), a i <b
numbers
i , i = 1, 2, . . . thesquncofpair mcble
a„ i=1, 2, . . .
.Itclearywibsufcentohwaesqunc
. Let A =A (e)
;intowclase
has density 0 .Wesplithquncea,
be
a p ~s A
a's forwhictexs
suficentlyarg
.Inthefirscla the
with a(a) :AP 0 (mod pA) . It follows from Lemma 2 that the density of the
a's oftheirscla 0
.
a(a) -0 (mod pA) for every p -A .
a(a) -ai -
For the a's ofthescndla
Thusclearyif
d
=b
i = 0 (mod d) .
A and d( a,., then a(a) -0 (mod d) . Henc
It follows from Lemma 3 thaexcpformst
Fx
of
the a's notexcdigwhave
(A
(4)
Or(a)
a,
(a)
ai
Since
a's of the
r canbehosritlyma suficetondrh
(4). Wehavfrom(4)nd thefac
secondlawhitsfy
d ~ A, d a ; implies db
(5)
;
(
b
b i
Now we have a(a) - a(b)=
)
a(a
' ) -r .
ai
' oio(a)
- ai
a i +b
a(a,)
a ;i
; . Thus from (5)
G(b)
b ;
or
(6)
-r.
1<b`
a.i
b i
a;
al
b i
Onamicbleurs
Hencfrom(6)ad
a(a;)
.
a ;±b
(7)
111
; we have
2 < c(a
a;
;) < 2
±r
Now it is well know') that the density of the integers satisfying a(n), n-c
existand coiusfntoc
. Thus it follows from (7) that the
densityofha' tescondla tifyg(4)soruficentlysma
r less than f', andsice
8
canbehosritlyma,heprof u
theormiscpld
.
REMARK . Lemma 2 together with Lemma 3 implies that for every
density of integers a, b forwhic
a(b)
a
a( )
is 0 for every F . Byamorecplitdagumenwcshotae
density of integers a, b
b
F .
b=
the
cr(a)-
forwhic
(jb)
is also 0 for every
- F,
~F
j(a)
a
Thusexcptforaqunedsity0
(j(b)
b
(RecivdJanury25,
a (a) ;
a
0( 1 )
1955.)
4 ) H, DAVENPORT, ÜberNumiAndates
. SitzungsberchdlPuBisen
Akademie der Wiss . 26 .'29 (1933), 830-837 . - Also : P. ERDÖS, Onthedsiyofabundt
numbers
. J. London Math .Soc
. 9 (1934), 278-282 . - P . ERDÖS, On the density of some
sequncofmbrs
. J. London Math .Soc
. 10 (1935), 120-125 .