SOME PROBLEMS ON
THE DISTRIBUTION
in this lecture I will talk about some
on the distribution of
recent questions
primes . It is not claimed that the pro-
blems I will discuss are necessarily important ones . I will mainly
speak about problems which have occupied me a great
deal in the
last few years .
Denote by
jf(X)
the number of primes not exceeding x . The
prime number theorem states that (X) X/
x -4 co . It
21
/
log X + 1
is well known that L A114knK gives a much better approximation
*As
to
1 0( ) than x/lob X . The statement that for X > X
is equivalent to the Riemann hypothesis . i n our lectures we will
problems like (1) but will rather consider problems of distribution of primes in the small (o . t problems on
not
.cal with
consecutive prime numbers) .
An old and very difficult problem on primp members requires
to find to every integer n a prime number p > n, or in other
words to be able to write down arbitrarily large prime numbers .
The largest prime number which is known is 2 2281 --1 , this number
numb: r by the electronic computer of the
Institute for Numerical Analysis the S.W.A.C. in the spring of
was proved to be a prime
1953 . It is an unsolved problem whether there ore infinitely
many primes of the form 2 p -1 .
1)
A few years ago Mills
proved that there exists a constant
n
c > 1 so that for all integers n, LLc")
is a prime . Unfortunately
it seems impossible to determine c explicitely . Very likely for
every a > 1 there are infinitely many integers n for which
is composite .
Put d =p
It immediately follows from the prime
c
-
P.Erdös
G -
number theorem that limd n/log n,;: 1 . Backlund ) proved that the
above limit is , 2 and Bracer and Zeitz 3) proved the t it is, 4.
Westzynthius 4) proved that liar . dn~lo ; n _ po . I proved ) using
Brun's method that for infinitemy many p.
6) proves+
and When
en
use of --run's method .
ly many n
the
same result much simpler without the
Finally Rankin7) proved that for infinite-
(2)
very hard to is prove (2) .
Ingham 8) proved that for n , n o do
It seems
ti on
tuned
In the opposite direcCraer9 ) conjec-
that
Using the Riemann hypothesis Cramer9 ) -proved that
1
I conjectured10
C Y"
that
s an immediate consequence of the prime
number
theorem, thus (3) if true is best possible . A simpler conjecture )
which I also can not prove is the following : Let n be any integer,
denote by a1 , a 2 t . . . a ' n) the integers
n which are, relatively
prime to n<. Then there: exists z constant C independent of n so
that
(4)
Anot her conjecture which seems very deep is that d o/log n has
T
V
"
a continuous distribution function i . e . that for every c 0 the
density f(c) of integers n for which do/log n < c exists and is
a continuous function of c, for which f(O)=O, f (cam )=1 . Hare is
another problem which has the same relation to the previous one
as (4) has to (3) : Put n,=2-3-5 . .-p
a 19(nk) arethe
k0lalyintegers, < n relatively prime to nk* Denote by A(c,n ) the
k
k
number of a's satisfying
Then lim A(c,n )
exi sts, and is a continuous
k
'I t follows from the trims number theorem that
in)
1, and I
proved that limn do log.; n 0 1 . It is
function of e .
lima
__ n /lopof course to
be expected that lim do/log No t in fact a well known ancient
11)
conjecture states that d =2 for infinitely many n .
n
proved that
I
further
I could not prove that
'YV
12)
proved that d o+1 , (1+ 1 )d n and d o+1 < (1- £ )d n
have both more than c I n solutions in integers m $ n . On would
Turan and 1
of course conjecture that l i
m d n" + 1 /J.-n n- oo and that lim d n+1 /d n=0
and that the density of integers n for which d o+1 > d is 1/2, but
n
these conjectures seem very difficult to prove . Rényi and 1 13)
proved that the number of solution of d
c .n/(log n)
3/2
solutionw is
dm , m n is less than
m+1
j the right order of magnitude for the number of
very
the equation d nvi
likely c .n/log n, but it is not even known that
=d
n has infinitely .many solutions . All the
P . Erdös
4 -
results mentioned hero arc o7 leaned by Bran's method .
In our paper with Turan 12) remark that we arc --- unable to
> d n+1 > d have infinitely many
n+2
n
are even unable to prove that at least one
prove that the inequalities d
solutions . In
fact we
of the inequalities d n+2 > d n+1 >d n y d n+2 < d o+1 < d n have infinitely many solutions
It se : ms certain that do/lob, n is everywhere dense in the
interval 'O,oo) . We prove the following
Theorem . " '
The set of limit points of d
form- abet
of positive measure .
Remark . Des nits the fact that the set of limit points of
do/log n has positive measure, I can not lecide about any given
number whether it actually belongs to our set, thus in particular
know if 1 is a limit point of the numbers d R/log n .
First of all it follows from the prime number theorem that
I do not
there are at least x/4 .1oz x values of
for which
4
'rk-l
(5)
We shall show that the
set of limit points S of dm/log
(or what is the same thing of dm/log x) of the m satisfying
m
(5)
have positive measure . (A point z is in S if there exists an
infinite sequence x1,--4 oo and m,_--j
00
theorem S
by a finite number of intervals the
If our Theorem. is false Khan .
can be covered
for
every
the Heine-Borel
lot (a
sum of the length of which is loss then
~K
,
be the intervals which. cover S . Let
k=1 1 2 9 . . . 9 1q 1
b Ja
i
0
I
(
. Clearly for all sufficiently
large x the d which
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the
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P . Erd6s
Kalmar 18) raised the problem Mether for every a>1 the density of
k
integers of the form p+ Ca 7 is positive . At present I can not
answer this question .
Following a question of Turn I proved 17) that if f(n) denotes
k then, lim f (n)=oo , in fact for
the number of solutions of n=P+2
that .
'
f(n)/loe n -iO ; but I can not even prove that there do not exist
infinitely many integers n so that for all 2 k~ n, n-2 k is a prime,
infinitely many n f(n) )
c .loglog n . One would guess
it seems that n=105 is the largest such integer.
Let a
< . . . be an infinite sequence of integers sati2
.
sfying a k 10 Ck
Denote again by f (n) the number of solutions of
n=p+a k* It seems likely that AT f(n)=oo .
1
< a
Van der Corput 19 ) and
. 1 18'')
proved that there exist infinitely many odd integers n not of the form 2 k +p q i n fact 1 18) proved
that there exists an arithmetic progression consisting only of even
numbers no term of which is of the
ve that for
k
form 2 +p . I also wanted to pr
every r there exists an arithmetic .progression no term
of which is of the form p+q
factors . This result
where q has not more than r prime
r
r
would easily follow if I could prove the
whilh seems interesting in
following conjecture on congruences
itself : To every constant c there exists a system of congruences
I
so that every integer
(8) .
satisfies at least one of the congruences
The simplest such system is O(mod 2), 0(mod
3) .
1(mod 4)
I
5(mod 5), 7(mod 12) .
Doan Swift constructed such a system with c=6, but the general
question seems vary difficult.
20)
Linnik
recently proved that here exists a number r so
kl+ . .
kr . It seems very
that every integer is of the form p +p +2
.+2
2
1
P . Erdös
7
likely that for ovary r there arc infinitely many integers not co
the form. p+2 + . . . +2 kr .
One final problem ; Is it true that to ovary c 1'02 and sufficiently large x there exist more than a 1 log x consecutive primes
2 x so that the difference between any two is greater than
21)1
c 2 2. If c 1 can be chosen sufficiently small this is well known .
F 0 O
T N 0
T E S
1) Bull.Amer.Math.Soc. 53, 604 (1947) .
2) Commentationes in honorem. Ernst Leonardi lindelőf Helsinki
1929 .
3) Sitz . Berliner Math . Ges .29, 1930, 116-125 . Sao also H.Zeits,
Elementare Betrachtungen űber eine zahlenteoretische Behauptung von Legendre .
4) Comm.Phys.-Math.9 Helsingfors, 5125/1931 .
5) Quarterly Journal of Math . Oxford Series , 1935, 124-128 .
6) Schriften des Math . Seminars and des Institutes fűr angewandte
Math. Der Univ . Borlin 4, 1938, 35-55 .
7) London Math.Soc. Journal, 13, 1938, 242-247 .
8) Quarterly journal of Math . Oxford Series 8, 1937, 255-266 .
9) Acta Arithmietica, 2, 1937, 23-45 ;
10) Duke Math. Journal 6, 1940, 438-441.
11) Publ.Math, 1, 1949, 3>37 .
12) Bull . Amer, Math .Soc . 54, 1948, 371-378 .
See further P.Erdös
.
Aid 54 9 1948, 885-889
13) Simon Stevin,Wis. Naturrk. Tijdschr . 27, 1950, 115-125 .
14) Professor Ricci informed me in the discussion following my
lecture, that this result is an easy consequence of some pre vious results of this . (Rivista di Mat.Univ.Rarma, 5 (1054)
3-54) .
15) E .Landau, die Goldbachsche Vermutung and der Schnirelmannsch
Satz, Göttinger Machrichten, 1930, 20-276 . In fact Schnurel
mann proves that the number
if
solutions of p 1 -p2 =
p1 <
8
P . Erdös
2 It
is loss than c x/(logx)
'0"(1+
16)
109 (1934) 668-678 .
17) Summa Brasiliensis Mathematicae, II (1950) 113-123 .
18) Oral Communication .
19) Van der Corput, Simon Stevin, 27 (1900) 99-105,(in Dutch) .
.Lapok 111 (1952) 122-128 .
P.Erdös (in Hungarian) Math
20) Trudy Math . Inst .Stoklov, 38, 1951, 152-09 .
21) P.Erdös, Acta Szeged, 15 (1949) y 57-63 . Soo also M . Cugiani
Annali di Matematica Sorie IV, 38 (1955), 309-320 .