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Sierpinski W. Elementary theory of numbers (Warszawa, 1964)(L)(T)(224s).djvu |
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for any integer a and p = 3, 5, 13...
For, if n divided by S leaves the
remainder r with 0 < r < S, then » = hB + r, whence 2"— 1 = 2M2r- 1...
Then there exist та+l numbers %,sa, ...,ccn+l
which are different roots of congruence C7)...
(To be more precise: и is the re-
remainder obtained by dividing со by a, v is the remainder obtained by
dividing m by *.) It is easy to verify that different pairs u, v correspond
to different roots of congruence D0)...
As regards the equation p(»-j-2)
= q>(ri), we know that for та < 10000 it has 80 solutions (for n <100
these are та = 4, 7, 8, 10, 26, 32, 70, 74)...
Since <p{l) — q>B) =1, the equation cp{as) =да, m being odd, is
solvable only in the case where m = 1...
Hence
I (p+l),
and so
Let us mention the following faet: there exists a natural number it > 1 such
that <f[n— l)/?>(«) > я» and g>(it+l)/<p(n) > m, and simuarly there exists a natural
number »> 1 such that р(л)/р(»— 1) > »» and у (»)/<?(»+1) > m (cf...
But, eince (j>, it) = 1, we have <р(рЪ)
= p(p)p(*) =(p- l)p<i) = 9>((p-l)b), and so putting n - (p— 1I;, we obtain
ifin + Tc) = <p(n), as required (cf...
It is easy to prove that if n — g?1 ??••¦?? w tbe factorization of
the number n into prime factors, then
We also have
d\n
g prime
§ 3...
Thus we see that for any integer a the relation j« | am-^m)(a'p("^-l)
holds for every г =1, 2,..., b...
of Carmichael (and so absolutely pseudo-prime) it is necessary and suf-
sufficient that X(m) \m—l (cf...
For each natural divisor д of number p — 1 denote by f(8)
the number of those elements of sequence B3) which belong to exponent
6 with respect to modulus p...
Hence, by theorem 9, 8 \ Ted', which in virtue of the assumption {"k, b) = 1,
gives S 1 S', and this, by 5' < b, proves that 5' = д...
It can be proved that if и is a prime and m a natural number > 1,
then in order that every integer be an wth power residue foT the modulus
m it is necessary and sufficient that m be a product of different primes,
none of the form «7c+1 (where 7c is a natural пшпЬет) (cf...
According
to the definition of the index, we have
Hence, using properties I and 1П, we obtain
ind^o...
On the other hand, suppose that an integer g relatively prime to p is not a
primitive root of p...
We have tie following sequence of
equalities:
Ж = с„ + дЖ1, Ж1=о1+дЖ2, ..., Жт_1=ст_1+дЖт, Жш = ст...
Bepresentation uf natural numbers
2ВУ
number s the sum of tlie s-th powers of the digits (in the scale of g) of the num-
gn— 1
ъег m = is n and, moreover, if n > 1, we have m > n; if n = 1, then we
put m = g ¦
9...
Sow let %
be a rational number which is equal to an irreducible fraction 1/m and
suppose that the representation of a; as a decimal is of the form A3),
where an (n = 1, 2,...) are digits in the scale of g where g is an integer
> 1...
It can be proved that each positive real number has precisely one
representation in this form and that a sufficient and necessary condition
for as to be an irrational number is that lim ~kn = +oo (Sierpinski [3])...
For ft = 2 its validity follows from C); we have
¦P»
*'
Suppose that D) holds for ft = m, where 2 < m < та...
This means that of any two consecutive convex-gents to x, the second
gives a better approximation than the first, Formula A0) shows that
>0
<0
for even n,
for odd n,
which means that the even convergent are less than я, whereas the odd ones
are greater than x...
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