Download SOME PROBLEMS AND RESULTS ON CONSECUTIVE PRIMES 5

SOME PROBLEMS
AND RESULTS ON CONSECUTIVE
PRIMES
by P. ErdL)s (Syracuse) and A. Rknyi (Rudspest)
5 1. Let $1 = 2, +, = 3, $s e 5, +,, . . . ##, . . . denote the
sequence of primes. In what follows we shall be concerned with
some problems regarding difference and quotient of consecutive
primes. We put
1c= 2,3, . * . .
There are many unsolved problems regarding the sequences d,
and Q,,, and the subject is full of peculiar difficulties. For instance
it has been conjectured by many authors that d+, = 2 for infinitiveIy many values of H, i.e. that the sequence of “twin primes”:
3,5
5,7
Il.13
17,19
29,31
41,43
‘..
is infinite. Neither this, nor the more feeble conjecture that dn
does not tend to infinity, has been proved up to now. Moreover
at present we are unable to prove even that
In this direction
Z&t - an = 0 .
W+COh.C~
it has been proved by P. ERD& r) that
Recently R. A. RANKIN 4) proved, that for C we can choose #+,
and according to an oral communication of A. SELBERG he can
improve this result by choosing for C the value #. This is all we
know about the small values of &. In the opposite direction it
can be proved by the method of V~GGO BRUN that twin primes
and more generally pairs of consecutive primes the difference of
which is equal to a fixed (even) integer 2R, are rather “few”. More
precisely, we have the following result *): The number of solutions
of 48 = k, $,, 5 x does not exceed
** 1
(3)
*) From the prime number theorem it follows only
d
r; 1.
MI A
n--too 1% n
**) We denote by c,, c,, . . . positive constants, and by c a positive
constant which is not always the same.
116
It follows from (3) that the sequence d,, is not bounded. As a matter
of fact, this follows also from TCHEBYSHEFF’S
estimation ‘)
6 (x) = dz log sb, < c,x,
h.6%
and also from the trivial remark that no one of the consecutive
numbersti!+2,7a!+3,n!+4,...n!+-+zisprime,andthus
there are arbitrarily large “gaps” in the sequence of primes. From
TCHEBYSHEFF‘S
estimation, mentioned above, it follows also that
F
dn
1 According to the primes number theorem we can
sly”, log 15, z X’
choose c, = I + E for any E > 0, if x is sufficiently large and thus
d,
we obtain lim 2 1. In his paper &) where previous results
N-MO 1% h are quoted, P. ERD&
proved, that
dn
> 0.
log &J . log log $H
(1% 1% 1% %4*
This has been still improved by RANK~N 8, by adding in the
numerator of the denominator in (4) the factor loglogloglog fi=
(4)
nFa
Q 2, If the conjecture regarding twin primes would be proved, it
would follow that the sequence G?,,oscillates between 2 and arbitrary
large values, thus it would be neither monotonically
increasing
nor decreasing, from some point onwards. This result has been
proved without any hypothesis, by P. ERD~S and P. TURAN ‘).
They proved also that the sequence q,, is also neither monotonically
increasing nor decreasing. These results can be expressed also by
saying that the sequences p,, and log p, are neither convex nor
concave from some point onwards. A generalization of this result,
regarding the sequence log fin, has brcn given recently by A. RENYI *). His result formulated in a geometrical terminology,
runs as
follows: Let us consider a finite polygonal line situated in the
complex z-plane and having the consecutive vertices z,, z,, . . . zn.
The total curvature of this polygonal line is defined by
(5)
where arg 2 denotes the argument of the complex number 2, i.e.
if Z = r . &, r > 0; - 7~< v I; 7t, we have arg Z = v. NOW let
TZNdenote the polygonal line with the vertices 2%= n + i. log &I,
&+l SNandlet
GN d enote its total curvature, It is evident, that
117
if the sequence log #+,would be convex or concave from some point
onwards, Gn7would be bounded. Rut RBNYI proved that GN tends
to infinity, as a matter of fact he proved GN > c. logloglog N,
thus giving a new proof of the theorem of ERI~S and TURAN
mentioned above. In waht follows we shall prove a refinement of
this result, giving the exact order of magnitude of GN:
TIteorem 1. If GIV denotes the total curvature, defined by (5),
of the polygonal line WV having the vertices zr, = it + i. log &,
p,+l 5 N, we have
cg . log N < GN < c, , log N.
(6)
lim - GN
exists, but we are at prcscnt
N i-CDlogN
not able to prove this, The pro01 of Theorem 1 will be completely
elementary. Besides the estimations of TCHEBYSHEFF
It seems probable
(‘1
that
*)
c -5
< 72(x) < cg -q
log f
410gx
and #,&+I < 2p,,, WC shall use only the method of BRUN.
5 3. Proof of Theomn 1. Clearly we have
(8)
Using
GN =+ +4; N 1arc tg log qH+l -----arc tg log
qn I.
n
this gives
arc tg a -- arc tg b = arc tg G
log %!k!
q*
X
arc tg
qn+1
. log pn ’
1
+
log
&+I 6 N
As according to (7) the sequence qn is bounded, we obtain from (9)
(f-9
GN=
(10)
where
C,LN<GN<LN
As regards the upper estimation,
it can be finished at once: we have
Now, we obtain
*) n (x) denotes the number
of primes
which
are
$ x.
118
Thus the upper estimation of (6) is proved. As regards the lower
estimation it is somewhat more intricate. We proceed as follows:
us neglect the terms
log y
of LN for those indices it for
I
I
which either 4, or c&,+1exceeds I Iog A,*) and denote the remaining
sum by LN’; evidently GN > c LN > c LN’. As according to (7)
Let
p$further
1 ) <I,
andwehavefor~xJ<l)log(l+x))>c,Ixl,
using the identity
9n-b1
-=
(14
* +a+14
9%
we obtain,
k4lrtn
--p-
ha
by putting
(where 2:’ denotes that the summation is extended only over
such indices n for which both dn and &+I are < 1 log fi) we obtain
LN > c7 (DN -
(16)
RN)
Now we have
Thus if we prove DN 2 c8 log N, GN > E%log N follows. To estimate
DN we need the following lemma.
Lemwuz I. The number of solutions of C = a, dn+r = b, P, < N,
does not exceed
In other words, the number of such primes p I; N for which both
p + a and 9 + a + b are also primes, does not exceed (18). This
lemma can be proved easily by the method of VIGGO BRUN (l.c.)
by applying a ,,triple sieve”. We need further
Lemma 2. Let us denote
w (4 =b,( 1+ f).
(19
----.
*) The
value
of the constant
.A shall
be determined
later
on.
119
We have
c
(20)
W(m) $;B+log(A
ASmsA+B
Proof of Lmma
+ B) + I.
2. Clearly
2
ASm6A-bB
ASdr5;
A+B
and thus
ASmSA+B
#?Y w (m)5
Now we prove, using Lemmas I and 2, the following
Lemmu 3. Let E, I and M denote positive constants, and let. k
denote a positive integer, k 1 kO(M, c, A), The number of indices S,
for which cl, < IM . (k -- I), k+i < XM (k -- - I) further
eMk
are valid,
does not exceed
cm. E . eMk
Mk
’
Proof of Lemm 3. The number of primes P, for which
CM(A--1) < p,, < eMk, further G!,,= a and &+I = b can be estimated
by Lemma I. If this number is denoted by &,b, we have
C@Mk W
Zasb-= @jp
(a) W (a + b).
If Zb denotes the number of primes satisfying
Lemma 3, we obtain
all conditions
of
for R > k, (M, E, il).
Let us proceed now to the proof of our theorem. According
(7) if we choose
to
(22)
Zk <
2
a<AM(R-l),
&,b.
b<hM(k-1)
lb-al < 7
Using Lemma 2 we obtain
120
C&Mk
we have at least -2Mk primes in the interval(eM(k-11, cMR),As we have
dN < eMk the number of primes P, in (@(k-1), e”)
,Wk-1) :b<&fk
,$A
does not exceed AM (k--l)’ and we have
the same estimate for the number of primes P, in the same interval for which dn+l 2 ;uM (R - 1). Thus for at least
for which d, 2 AM (k--l),
&k), WC have both dn 5 UM (k--l)
and
primes in (&P-i),
dpl+l < AM (k-l).
According to (23) the number of primes in
(#(k-l), eMk) for which di < AM (k--l), &+I < AM (k-l)
and
&kM does not exceed ‘g,
Let us choose now R=
p*+l--&tI
< --j=-
32
c
and E = &.
c eMk I
.
We obtain, that for at least kk
primes m
(eMF:-11, eMk) we zve
EkM
jdn+wZ+T.
d, < ctM (k-i),
&+r < AM (k-l),
and
Thus from (15) we obtain
1da-t1 --dn I> 5
2s’
(24) DN > 2C
4ilM
P*
kD<k < W N ,Wk-1 1<p,,<$fk
-F
As it has been remarked
L:
I >c,, 1ogN.
k,<k<‘%$!
above, this proves our theorem.
5 3. In connection with the above estimations there arises the
question, how many times it occurs for P, 5 N that d~+i = dn,
or more generally that &+l = 4, + h, where h is a fixed integer
(positive, negative or zero}. In this direction we prove
Theorem 2. The number of solutions of $+, I; N, u&+-r = & -t- h
does not exceed
(25)
Proof of theown 2. We have according to Lemma 1
121
Applying
the inequality
of
we obtain
CAUCHY-SCHWARZ,
HN<- ca
w (m).
logs%s, (logs)%+*
(27)
Row we need the following
simple
Lemma 4.
,#;I wa W < crs-4.
(28)
Proo/ 01 Lenama 4. We have
I e
1
n”
==z If
p=a (
P=S 1 + p
(
P)
( I+$
1
pr+
2$ 1 =c11
and thus
where V {d) is the number of different prime factors of d. Denoting
by G (d) the number of divisors of d we have
and thus
Thus we obtain
$Ws(m)
m-l
<$A
which proves Lemma 4, As &
(log IV)% can evidently occur
4:
not more than N (log iV)-‘h times, as it is seen from 2 dn 5 N,
ASN
Theorem 2 is proved.
$ 4. We prove still some results concerning the sequence dn.
Recently W. SIERPINSM
proved the following theorem: *)
For every positive integer K there exist an infinity of primes +
with the property that all the numbers $ -f R, k = 1,2, . . . K are
composite i.e. that there are an infinity
of primes which are
“isolated” from both sides from the next prime by an interval of
arbitrary large length. Evidently
the theorem of SIERPINSKI
is
=
0.
The proof given by
SIERPINSKI
(‘7
122
is based on the application of the theorem of DIRICNLET,
that in
every arithmetic progression Dx + Y, x = 1, 2, . . . D,Y = I there
are an infinity of primes. In what follows we shall show that a
theorem which is stronger than that of SIERPRINSKI
can be proved
in an elementary way by using only BRUN’S method. We shall
prove
Theorem 3. For any integer N and any Y < &log
prime &, I; N for which
N there is a
1% N j==O,1,2,,..(r-I).
dH-i 2 Cl6
_____ra
'
Proof of theorem 3. We need the following
Lemma 5.
2 W (m) <
3A1-A
for O<A<
(I -a) 21
m=l d
(291
1.
We have namely
which proves Lemma 5. Now WC Provo
Lemma 6.
(30)
We apply the theorem of Vrcco
We obtain using also (7)
I
(31)
p$
a
5&N
hSN
-’ (,0&A
RERUN
mentioned in $ 1 (see (3)).
i+
anA
n (N) ’ ,,&
,:g’N
W (4
’7
+ (lo;N:
+A’
Thus by Lemma 5 WC obtain from (31)
(32)
1 r; 3c N (log N)r-A
-A)
pl:N ad - logsNil{
= qi--
3c N --a) (log N)‘+A’
which proves Lemma 6. Now Theorem 3 can be deduced easily.
Let us put
123
T-l
1
c
j=l dG’
We have by Lemma 6 with ,I= 4.
8(‘)
n =
(33)
c
hsN-r-i-1
(34)
Thus if
we
put
(7) that
Atin
8:) = ,fi’,) we obtain
hS N-r+1
(35)
A:r’ < ;gN*
Thus for an appropriate
it we have
(36)
dw-j
>
Cl6 1% N
Y3
from (34) using
for i=O,l,...(r-.-1),
which proves Theorem 3. For Y = 2 this thcorrm states, that there
exists a constant C such that for every N there is a prime fi 5 N
for which all the numbers 9 f k, k = 1, 2, , . [Clog N] are
composite. This can be formulated also by saying, that the first
prime having the propcrty required by the theorem of SIERPINSKI,
i.e. for which all numbers + f k, k = 1, 2, . . I K are composite,
does not exceed eCK.
We still add some remarks on Lemma 6. As it has been used
CN
before, for more than primes & 5 N we have d, < K log N,
log N
and thus we obtain
(37)
This means that the order of magnitude of the sum on the left of
(30) given by Lemma 6 is exact. Now we can make the parameter A
in this sum depend on N, for example we may choose
A=l.--
1 .
loglog N’
We obtain from Lemma 6 that
(38)
cl,Nlogrog
N
c ‘<
p*SN
46 logsN--’
For the sum on the left of (38) we can give only the lower estimate
(39)
ArN’
1
z fi ’
%l!DN
(log N)”
124
As amatter of fact, if we could prove
a%
that dim = 0, but at
n-cc0 logfi
present we ate unable to prove anything of this type. To give some
idea for the random distribution of the sequence A, we add a tabel
giving the values of ~6, for P, S, 4397, (n = 2,3, . . . 599).
with
y(N)
Budapest,
-+ 00, it wou1.d follow
January
30, 1949.
References:
1. P. EIZDOS, The
(1940) 448-441.
2. R. A. RANKIN,
4.
5.
6.
7.
8.
0.
of consecutive
primes,
Dnke
Math.
Journal,
The difference between consecutive
prime numbera,
III,
Sot. Journal,
22 (1947) 228230.
VIGGO BRUN, Le crible
d’Eratosthene
et le th&oreme
de Goldbach,
Videnskapselskapets
Skrifter,
I, No. 3, Kristiania,
1920.
P. L. TCHEBYSHEFF,
Memoire
sur les nombres
premiers,
Journal de
math. pures et appl. I. ser. 17, (1862) SW-3!l0
pp.
P. ERD&,
On the difference
of consecutive
primes, Quarterly
Journal
of Mathematics,
Oxford ser. 0, No. 22, (1936) 124-128 pp.
R. A. RANKIN,
The difference
between consecutive
prime
numbers,
London
Math. !%oc. Journal,
13 (1938) 242-247.
P. ERD~S and P. TuRhN,
On some new question
on the distribution
of prime numbers, Bulletin
Amer. Math. Sot. 54, No. 4, 1948.371-378
pp.
A. R~NYI,
On a theorem
of Erdiis and Turan,
Bulletin
Amer. Math.
Sot. in print.
W. SIERPINSKI,
Remarque
sur la repartition
dea nombres
premiers,
Colloquium
Mathematicum
1 (1948) 193-194.
London
3.
difference
Math.
125
n
0 t
10
238
40
50
00
70
iFi
100
IO
28
30
40
50
60
70
80
on
300
10
20
30
40
60
i;a
;;i
6a
300
IO
20
3a
40
60
(iC
TC
80
4z
16
2c
3c
41
6C
Ii0
70
80
Table of the sequence d, for IZ = 2,3, , I . 599.
0,
1,
2,
3,
4,
6,
6,
7,
I
0
4
19
Q
2
4
2
8
12s
2
0
:
2
lb
4
6
i
u
1:
10s
t
4
4
t
4
4
4
2;
u
2
0
1;
240
10
:::
1:
10
:
u
1:
8
1:
0
1:
2
6
6
8
1;
14
1:
10
2
a
2”
10
12
12
i
:
12
12
1:
2
8
II
14
I3
4
2
11
;
12
t
12
4
6
(i
14
1:
1:
11
12
12
$
20
30
40
60
60
1:
18
22
2
Rcl
ii
12
8
ii
t
-
4
4
8
10
2
a
E
4
;
12
20
1::
10
2e
1:
4
t
2
8
i
28
4
0
2
6
:
4
t
1:
12
4
:
:
4
2:
2:
:28
12
8
1:
2:
14
2
1:
11
I8
4
2
2
4
2
1;
ii
8
1::
la8
6
8
1:
i
1:
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1;
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6
2
2
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1:
4
:
4
6
1:
1:
;:
1:
2
1:
;
:
1:
1:
2
1:
6
6
:
1;
1:
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8
4
14
1:
14
2
8
11
2
0
0
4
6
2
:
10
14
t
1:
4
4
2
6
tl
1:
t
10
6
:
2
8
1:
2
2
4
0
6
2
6
4
I(i
6
2
8
I6
6
4
t
6
4
2
2;
4
2
2
4
4
1:
1:
10
18
2
2
G
1:
8
0
1:
4
t
*t
ii
4
1:
;
6
6
1:
IO
2
2
2:
2
8
:f
10
D
3::
8
2
6
4
::
:
1:
10
4
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1:
6
8
“6
8
1;
:t
8, -2
1:
II:
12
4
:t
0
t
4
4
6
:::
4”
2
20
6
6
0
4
f
122
10s
1:
lb
10
2
1:
II
1:
12
6
fi
2
2
4
4
i
6
4
II
2
4
22
1:
24
4
0
8;
2
1:
4
8
1;
2
6
2:
E
i
22
10
8
11
8
2
1:
11
:
4
4
22
2
i
10
1:
18
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8
f
20
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I:,
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0