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Sierpinski W. Элементарная теория чисел (Warszawa, 1964)

Sierpinski W. Elementary theory of numbers (Warszawa, 1964)(L)(T)(224s).djvu

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Date Jan 16, 2005

Cites: We have already mentioned Beeger's theorem that there exist infinitely many
even Poulet numbers...
This proves that the congruence g(xt)
= O(modj)) has at least n different roots, which contradicts the assump-
assumption that theorem 13 holds for polynomials of degree n—1...
Then, in virtue of the identity
4a(ax*+bx+c) = Bos+*J— (ft2— 4ac),
congruence D4) can be rewritten in the form
D5) Baa;-|-uJ=D(mod4am)...
Since D is odd, the numbers z and «0 are also odd, whence it follows that
the numbers я—г0 and s+za are even...
Let m be a natural number, I a natural number relatively prime
to in, and r an arbitrary integer...
Therefore the total number of the numbers of the
table, which are relatively prime to m and to I, is cp(l) rp(m)...
Erdos [18], if
for a given natural number s there exists a natural number m such
that the equation <p (n) = m has precisely s solutions (in natural num-
numbers n), then there exist infinitely many natural numbers m with this
property...
The sum of the values of Uuler's totient funetimi over
the set of natural divisors of a natural number n is equal to n...
It follows immediately from theorem 8 that for any natural number
in there exists the least natural number X (m) such that in [ <j*'m' — 1 for
(a,ro) =1A)...
If q = 3, then 2-1 =¦ 0(mod9), whence 22p = I(mod9) and, by theorem 9,
I
248
СЯАРТЕВ VI...
Since p \ nn—1, -we
5
have 5 | n, whence we infer that nj5 is a natural number >1 (because
6 < n)...
As we learned in § 6, there exists an integer Ь such
that 0 < A <2>—2 and a = ^(modp) which, in...
In this way we find 2 = 2, 22 = 4, 23 = 8, 2" = 3, 2s = 6, 2s =12,
2' = 11, 2s = 9, 2" = 5, 210 = 10, 2" = 7, 213 = I(modl3)...
But since
p\g2 —1 is impossible (because g is a primitive root of p), p\g2 -r 1 is
valid, i.e...
it can be rewritten
in form A), where the numbers en (n = 0,1, ..., m) are integers which
satisfy inequalities B)...
Considei an arbitrary infinite sequence c1; c2,..., wheie en (n — 1,2,...)
are digits in the scale of g...
(If, more-
moreover, the number is irrational, there are at least two digits that appear infinitely
many times each.) However, for numbers V2 and tz we are unable to establish which
two of the digits have this property...
Hence
Z-Vo' _ 1^11 1
2 ~? ' 3^7 3-7-47" 3-7-47-220T T ""
This expansion is to be found under the name of Pell's series in a book
by E...
Applying the above-mentioned
argument slightly modified, we conclude that if numbers Bk are defined
by B), then for any natural numbers n, m > n, we have
--BJ <
«(n+l)
This proves that the infinite sequence Mn (» = 1, 2,...) is convergent,
i.e...
In fact, we have ax — [jKjJ = I 1; therefore in order that
Lx— a0J
% = m it is necessary and sufficient that m <l/(«— «„) < m+1, i.e...
Sow we are going to find these natural numbers for which the period
of the representation of ^D as a simple continued fraction consists of three
terms...
An argument similar to that used in the previous case shows that
number D5) is the (Ъ — l)-th convergent of the simple continued fraction
fqr number \/jD and that It = an, where s is the number of the terms
of the (least) period of the continued fraction for vD and n is a natural
number...
Hence
(8)
the congruence being valid since, in view of the theorem of Fermat,
D*-1 =i (modi))...
Prom this we infer that the
number of pairs which belong to the first class is У — • Similarly,
ii L 1 J
the number of the pairs that belong to the second class is
is ? ["Г J ¦
Siace
Elementary theory of numbers
322 CHAPTER IX...



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