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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: Thus we see
that conjecture H implies the existence of infinitely many absolutely
pseudoprime numbers...
(It can be proved that the numbers 10, 26, 34 and 50 are not of this
form.) We do not know whether every odd number is of this form...
As follows easily from theorem 3, for any natural number n we
have ф(пг) = mp{n), consequently <p(a?) = aip(a), tp(b2) — btp(b), whence tpC>):<p(y)
= a: 6, as required...
In order that the greatest common divisor of the numbers m < n and
n be й it is necessary and sufficient that m = Jed, where Jc is a natural
number <n/$ relatively prime to nju...
For as > 1 formula A0) gives
where ? denotes the sum extended all over the pairs of natural numbers
А-Л such that W<a...
On the other hand, it is clear that congruence A9) has natu-
natural solutions only in the case where (a,m) =1...
It is also plain that without loss of generality we may suppose that
n > 2 (for in the sequence of odd numbers there exist (as we know) infi-
infinitely many primes)...
In other words, it has turned out that numbers rk of
sequence B6) which belong to the exponent 8 with respect to the modu-
modulus p are precisely those whose indices Ъ are relatively prime to 8...
Thus we see that the assumption that a number m
is a product of more than four prime factors leads to a contradiction...
According to the definition,
i = (p—X,n), which, by theorem 16 from Chapter I, proves that there
exist two natural numbers я, v such that d = nu — (p — l)u, whence
id = Tcnu — k(p—l)v...
Consequently, number Ж has at least one prime divisor p — 67c —1,
where Ъ is a natural number...
It can he proved that the least natural number > 9 with this property is the
number 8712 = 4-2178 and that the numbers written above exhaust the class of
the numbers of this property...
For g = Y2 and ж = B/2 +l)/4 we have two representations in the
form A3):
2l/2 + l
the latter being given by the algorithm...
In fact, if the decimal of a were recurring, then,
since all numbers 10" (» = 1, 2,...) occur in it, arbitrarily long sequen-
sequences consisting of 0's would appear; consequently, the period would neces-
necessarily consist of number 0 only...
Therefore, in virtue of the relation xn = an+l/xn+1 and C1), we in-
infer that а%_! = ^^„i...
Hence, in particular, for t = X, we obtain
^12 = C; 2~7б), V^il = F; 2,2,12),
/180 = A3; 2,2,2,26), ^926= f30; 2,2,2,2,60)...
To close this chapter we consider the following continued fraction
~T "
Let ft,, 6a, .....
By A3), these two formulae show that the formula
is valid for any two different odd primes p and q...
This is obtained immediately from theorem 5 by a simple applica-
application of the identity
and by the remark that for p =3m1+yi we have the equality 3a>2±
±2xy—y2 =p—2y1±2soy the right-hand side of which is different from
zero since p = 64+1 is odd...
First of all we note that the task of calculating the value of l-^-l,
where P is an odd number > 1 and D an integer relatively prime to P,
may be reduced to that of calculating the value of l-^)i where Q is an
odd natural number...
To calculate I—I we look at equalities B1) and
find the number m of the pairs Pi_x and, eJPi in which both P^_i and BiPt
are of the form 44+3...
We now prove a theorem which, in a number of cases, enables us
to decide whether a Mersenne number is composite or not...
Neither are we able
to prove that there exist infinitely many square-free Mersenne numbers...

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