Download pseudoprime or a Carmichael number

201
On pseudoprimes and Carmichael numbers .
Dedicated to the memory of my friend Tibor Szele .
By P . ERDÖS in Haifa .
A number n is said to be a pseudoprime if
(1)
2"=2 (mod n).
It is said to be an absolute pseudoprime or a Carmichael number if for every
(a, n) =1
(2)
a ,, -a (mod n) .
Denote by P(x) the number of pseudoprimes and by C(x) the number of
Carmichael numbers not exceeding x . It is known that')
(3)
c, log x < P(x) < x exp (-c.>(log x) 4 ) .
KNÖDEL ") recently proved that
(4)
C(x) < x exp (-c„(log x lOg log x) 2 ),
, as x
it is not yet known whether C(x)
i . e . it is not known if
there are infinitely many Carmichael numbers . In the present paper I prove
by KNÖDEL'S method that
(5)
and
P(x) < x exp (-c4 (Iog x log log x) )
(6)
C(x) < x exp (-c3 log x log log log X./log log X) .
for a suitable positive o . I would rather
conjecture that C(x) > xl for every s > 0 and x > x(8), in fact I believe that (6)
can not be very much improved . I shall give some heuristic reasons for this
guess . Finally I shall state some theorems without proof .
As far as I know D . H. LEHMER 1) was the first to discover that there
are even numbers n which satisfy (1), and BEEGER 3) proved that there are
KNÖDEL conjectured that
C(x) < x' - "
-E
1) P . ERDÖS, Amer. Math. Monthly 57 (1950), 404-407.
°) Archiv der Math . 4 (1953), 282-284.
s) Amer. Math. Monthly 58 (1951), 553-555.
202
P . Erdös
infinitely many such integers . I do not know if there are any composite
numbers n for which cr' -a (mod n) for every integer a (i . e . not only for
the (a, n) = I).*)
Throughout this paper c,, c, . . . will denote positive absolute constants,
pi and P,, will denote primes, log, ;x will denote the k times iterated logarithm .
First we prove the following
Lemma 1 . Denote by N(p, pp;. ; x) the number of integers not
exceeding x composed of p,, p, . . . , p, ; . Put k" = x. Then for u < log x log . x
(i. e. k > log x)
N(p, p, . . ., p,: ; x) < x exp (-c,, u log u) .
Clearly N(p,, ,vp,, ; x)- N(2, 3, . . . . ;
P,, x) where P,: denotes the
k-th prime . Now -r(k) > k. Thus by a theorem of DE BRUIJN 4)
N(p, p, . . ., pr: ;
x) -- N(2, 3, . . . ; P,: ; x) < tp(x, h~') < xexp (-c, ; U log
11),
x all whose prime factors
where 1'(x, y) denotes the number of integers
.
Thus
our
Lemma
is
proved.
are
y
Now we prove (5) . Denote by l2 (p) the smallest exponent satisfying
2'20" -- I (mod p) . We split the pseudoprimes nOt exceeding x into two classes . In the first class are the pseudoprimes n every prime factor of which
satisfies
:_E~;
1.2 (p) < exp ((log x .log.,x)-).
Clearly all the pseudoprimes of the first class are composed of the prime
factors of
t
(7)
2`-1, 1 <t<exp((logx ' log.2 x)'<
The number of prime factors of 2'-1 is clearly less than t, thus the number k of prime factors of all the numbers (7) clearly satisfies
I
k <
t2 < exp (2(log x log.
;X)")
Thus by Lemma I the number of pseudoprimes of the first class is less than
(8)
xexp (-c, (log x log.2 x) ) .
The u of Lemma I here equals c .,(log xx log2 x)= .
Every pseudoprime of the second class has a prime factor p satisfying
l .,(p) = exp ((log x 1092 X)2) . Since
(9)
n--- O(mod p),
n is a pseudoprime we must have
n=1 (mod 1_,(p)),
Added in proof : In a recent letter Dr.
ber has the above property .
4) Indag. Math. 13 (1951), 50-60 .
KNÖDEL
n > p,
proved that every Carmichael num-
On pseudoprimes and Carmichael numbers .
(for if n =p, n would not be a pseudoprime). Thus
n > p - 12 (p) .
203
Let now
x0-
p„ p2 , . . ., p,, be the primes not exceeding x for which 1.,(p) _ exp ((log x lOg-2
We have by (9) that the number of pseudoprimes Of the second class does
not exceed
{10)
x 1` x exp (-(log x .1og_x) ')
- 1 p ; l., (P ; )
xexp(- 2 (log x
1
.log.,x)-) .
(8) and (10) clearly imply (5) .
Now we prove (6) . Let k be an integer and denote by f(k) the least
common multiple of p ;-1, j= 1, 2, . . . where p ; runs through all the prime
factors of, k . First we state
Lemma
2.
The number of solutions of f(k)
y exp (-c„ logy •
t, k .~- y does not exceed
log : ;y log-2y)
(independently of t!)
Let us assume that Lemma 2 has already been proved . Then the proof of
(6) proceeds as follows: It is well known (and obvious) that n is a Carmichael number if and only if it is a composite, squarefree number such that
for every prime factor q of n, q-1 divides n- I . We split the Carmichael
numbers not exceeding n into two classes . In the first class are the Car,
michael numbers whose greatest prime factor is greater than x'U . Let n be
a Carmichael number of the first class and p its largest prime factor . We
evidently have
n - 0 (mod p), nI (mod (p-1) ),
n > p.
Thus as in (10) we have that the number of Carmichael numbers of the first
class is less than
{11)
x
.
p(p1-1) G X"
Let now n be a Carmichael number of the second class . Write
t
n --PIP, . . ., p,: ,
x'' = p, > P, > . . . > p,:k
Assume n > x' . Define
pl,u, . .,
p;
,'x'
PIP, .
.
.,p;=x'
<n .
Now
(12)
n-0(mod p, . . .p ;),
n- =1(mod f(p, . . .p,)), n>p, . . .p;
(i . e . n - I (mod (p ;-1)), 1 -j- i) .
204
P. Erdös
,
2
Let k be any integer satisfying xi < k _- x 3', . We have from (12) (as in (10))
that the number of Carmichael numbers of the second class is less than
kf(k) ,
x m x 1~
1
where the dash indicates that x 2 < k ~ x 1 . Now we have to estimate
(14)
where in `i
(15)
- 1
', kf(k)
=- .-',
f (k) > exp (c, ;, log x log, x, log.) x) . Clearly
i < exp (-c,,, log x log :; x log., x)
1
5
< exp (-c,,, : 2 • log x log:3 x
k
log1 x) .
,:
Next we estimate
v_.
We have by Lemma 2 that the number of k < y (y > x 1 )
satisfying f(k) < exp (c,,, log x log,x, log =x) is for sufficiently small c,,, less than
(16)
y - exp (-c„ logy log:;y log.,y)
• exp (c,,, log x 1093X. 1092X)
<
< y • exp (-c„ log x log :;x log .1 x) .
Thus from (16) we have
(17)
`.; kf(k) < ~~
.;
k<
exp (-
exp (-c„ log x log-X , log-..X)
z
2-
k <
log x log,x ; log 2 x ` .
From (13), (14), 15) and (17) we have that the number of Carmichael numbers of the second class is less than
(18)
x • exp (-c,, log x log :;x log:, x) .
(11) and (18) prove (6)
Thus we only have to prove Lemma 2 . f(k)-t implies that all prime
factors p Of k satisfy (p-1) t. DenOte by p,,p,, . .,
all primes for which
(p-1) t . Then the number of solutions of f(k) - t, k - y is clearly not greater than the number of integers -_ y composed of the primes P,, P2 , . . . .
Denote by q„ q.,, . . .
exceeding
the primes
among
the
p i 's
not
exp ((Iog1 y)',-logy) and by r, < r, < . . . the primes among the pi's greater than
exp((log .,y)/log:;y) . Put k- Q. R where Q is composed entirely of the q's
and R is composed entirely of the r's . Consider
1
f(k)=t,
1
Then clearly either Q
> y4
y 2 <k-y .
1
or R > y 4 .
Thus the number of solutions of
On pseudoprimes and Carmichael numbers .
20 5
1
f(k) = t
is clearly not greater than (the dashes indicate that
y 4 <R<y)
y
4
I
IV,
(19)
3' ' LY~
RI- ~
-
Q ~
r-
From Lemma I we have that the number of integers Q < z,
is less than
z•
(20)
exp(-c, : ;
y4 < z < y
log y Iog3 Y. log.,y) .
(The u of Lemma I here equals c„ logy log, y'(log.,y)"-) . Hence by (20)
(21)
~~ < exp (-c„ log y log;,y logy)
17,
1
< ex P (- c,,1 2 • tog y loglog=j3•'
g2 Y) •
. The is are the primes - y greater than
k
which satisfy (r;-1) t. First we show that (P.; is the
Now we estimate `'
exp • ((log :,j,)-/log; ;y)
i-th prime)
r; > (2i log i) 1- ~ > P' - a,
(22)
rc = c,-,
logy log., y .
Let s,, s, . . ., s ; be all the prime factors of t . Then the number of is
not exceeding r ; (which of course equals i) is not greater than N(s,, s:,, . . ., s; ; r,:) .
Now -') t < k -- y thus j < logy . Put r ; _ (log y)"" . Then we have as in the
proof of Lemma I by the Theorem of DE BRUIJN 4) (, c((log y)2) > log y)
(23)
i -- N(s, s, . . ., s; ; (log y)"') < J,((log Y)'{", (log y)")
< (log y) exp(-c,, ; u ; log u ;)
(r ; = y, thus Lemma I applies) .
Now r ; > exp((log,y)','log;y), thus u ; > 1og .2 y'log ;;y, hence (22) follows
from (23) by a simple computation .
Now R„ R,, . . . are the integers composed of the r's . We have by (22)
. Thus
R ; > i1-a
(24)
";"1
6' ;
1
where in `, R ;
dently have
> y
-1
4,
Y
1
may be empty .) We evi-
yg and in `.- i > y 8
1
Y+
( 25)
a,
1
i =
1 --1
1, ,
~, <
ti,
i'1 «
< c," (~r U
J,
exp(-c logy log: ;Y' log_y) .
Namely t - f (k)
g (/") L
8)
1
206
P . Erdös : On pseudoprimes and Carmichael numbers .
From (24) and (25)
1- exlogtolog
- R
( 226'
6)
P
(-C,-,
a
1
g:. :y ~ ._y) .
(19), (21) and (26) complete the proof of Lemma 2 .
Let now A - = p : p 2 . . . p,, be the product of the consecutive primes less than
s log x, and denote by r,, r_, . . . the primes for which (r;-1) A . It is easy
to see that A x' , for sufficiently large x . It is reasonable to expect that for
ii (log x)` " there are more than c ;,, : -r(u) (c,,,== c,,,(c,..)) is not exceeding u
(though this will probably be very hard to prove). If the preceding statement
is true, then a simple computation shows that there are more than x" composite, squarefree integers n
x composed entirely of the i's . Again it is
reasonable to assume that these numbers are roughly equidistributed (mod A),
and thus one can assume that there are more that x' -1 ' composite squarefree
integers less than x -- I (mod A) which are all composed of the r's .
Let n be such a number, then n is clearly a pseudoprime, since all
prime factors of n are r's, n - - I (mod A), (r ;- 1) A . Thus, if all the above
conjectures are true, log (C(x)) . log x, 1 .
In a previous paper') 1 proved that for a suitable infinite sequence x,
the number of solutions of ff(n)--x ; is greater than x; where (1 :(11) stands for
the Euler-function . The abOve arguments (using only the first conjecture) would
imply that c,,, can be taken as close to I as we please . By arguments similar to
those used in proving (6) 1 can show that the number of solutions of (f (11)
x
is less than
x exp (- , log x • log;,x log-2 X) .
I can further show that for any r, 1 and x
X_
.,x)
log x (log
~.
x„(F, k)
x-
f(k) < 109 z (log x)'
.
and if one neglects a set of integers n ; of density 0 then for every
log n- (I + .) log,n log ;n _ log (f(n))
log n-(1
Thus, in particular, for almost all n and every c
f(n)==0
-- - n - -~
(10g 11)'
.
(Received September 5, 1955 .)
) Quarterly f. Oxford Ser . 6 (1935), 211-213 .
-F)
F
> 0
loga n log ;n .
INDEX
Tibor Szele and his mathematical life-work (A . Kertész)
115
Gáspár, J . Eine axiomatische Theorie der kubischen Determinanten 126
Obláth, R. Sur la répartition des nombres sans diviseur quadratique131
Pukánszky, L . Some examples of factors 135
Pickert, G . Eine nichtdesarguessche Ebene mit einem Korper als Koordinatenbereich . . 157
McCoy, N. H . The prime radical of a polynomial ring
161
Dénes, P. Über den zweiten Faktor der Klassenzahl and den Irregularitätsgrad der
irregulären Kreiskörper 163
Gyires, B . Eigenwerte verallgemeinerter Toeplitzscher Matrizen 171
180
Ellis, D . A theorem on description adequacy 184
Sierpinski, W. Sur une propriété de la fonction T(n)
Neumann, H. On the intersection of finitely generated free groups 186
Neumann, B . H. On some finite groups with trivial multiplicator 190
195
Surányi, J. On a problem of old Chinese mathematics Bateman, P. T. and Erdös, P. Partitions into primes
198
201
Erdös, P . On pseudoprimes and Carmichael numbers Moor, A . Allgemeine metrische Räume von skalarer Krümmung 207
229
Kertész, A . Modules and semi-simple rings . 11
Szász, F. On rings every subring of which is a multiple of the ring237
239
Iséki, K. On the Lebesgue property in uniform spaces Mordell, L . J. Diophantine inequalities in complex quadratic fields 242
Fejes Tóth, L . On the volume of a polyhedron in non-Euclidean spaces256
262
Steinfeld, O . Über die Quasiideale von Halbgruppen
Rapcsák, A . Über das vollständige System von Differentialinvarianten im regulären
276
Cartanschen Raum
Sods, Gy. Über Gruppen von Automorphismen in affinzusammenhängenden Räumen
294
von Linienelementen
303
Rédei, L. Die endlichen einstufig nichtnilpotenten Gruppen . .
325
Aczél, J. Über Additions- and Subtraktionstheoreme 334
Erdös, J. On the structure of ordered real vector spaces 344
Szép, J. Zur Theorie der Halbgruppen 347
Baer, R . Norm and hypernorm
Los, J., Sasiada, E . and Slominski, Z . On abelian groups with hereditarily generating
351
systems
357
Balcerzyk, S . Remark on a paper of S . Gacsályi Vranceanu, G . Sur les espaces á connexion affine localement euclidiens 359
Péter, R . Die beschränkt-rekursiven Funktionen and die Ackermannsche Majorisierungs362
methode
376
Fodor, G . On a problem in set theory Carlitz, L . The number of solutions of a particular equation in a finite field379
Barna, B . Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von
Wurzeln algebraischen Gleichungen II ,384
Erdös, P. and Rényi A . On some combinatorical problems 398
Turán, P . Remark on the zeros of characteristic equations 406
Prékopa, A . On the convergence of series of independent random variables410
Varga, O . Eine Charakterisierung der Kawaguchischen Räume metrischer Masse mit-
tels eines Satzes über derivierte Matrizen
418
Hajnal, A . and Kalmár, L . An elementary combinatorial theorem with an application
to axiomatic set theory
431
Szendrei, J. Zur Holomorphentheorie der Ringe 450
Higman, G. Complementation of abelian normal subgroups 455
Hosszú, M . On analytic half-groups of complex numbers
459
Kovács, L . A note on regular rings
465
Beaumont, R . A . Matric criteria for the uniqueness of basis number and the equivalence
of algebras over a ring
469
Egerváry, E. Auflösung eines homogenen linearen diophantischen Gleichungssystems
mit Hilfe von Projektormatrizen
481
Fried, E . and Varga, T. Über die Frage der Einführung der komplexen Zahlen . . . . 484
Fuchs, L . Ringe and ihre additive Gruppe
488
Bognor, M. Ein einfaches Beispiel direkt unzerlegbarer abelscher Gruppen509
K y i n m 0 B, 9 . 2 . 0 npsimhix pa3iioweHuax ojH01i cMemaHHou a6enesoii rpynnbl . .
512
IM, N. Über das Produkt von zwei zyklischen 2-Gruppen 517
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