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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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Cites: Then there exist та+l numbers %,sa, ...,ccn+l
which are different roots of congruence C7)...
In order to establish the converse correspondence, that is, to find
for a given root z of congruence D6) all the roots cc of D3) to which
the root я corresponds, we have to solve the congruence 2ax+b
= z(mod4a»i)...
According to the lemma, the remainders obtained by divid-
dividing tbe numbers of the column by m fill up the set 0,1, 2,..., те —1,
whence the number of the numbers of the column, which are relatively
prime to m, is <p(m)...
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)...
We do not know whether there exist infinitely many natural num-
numbers which are not of the form n—р(та) where и is a natural number...
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...
Thus we see that the equation if (n) = 2-36t+I, where 7; is
a natural number, has precisely two solutions, n = 36J:+I and n = 2-36i+2...
So tit2-..tB sa a3 i
1 (modr), whence,
in view of (a, r) = 1, by theorem 8, we infer that aPW ™ l(modr)...
Numhers which belong to a given exponent
247
Number 7 is also a primitive root of number 10 since 71 = 7, 72 = 9,
73 = 3, 74 = 1 (modlO)...
For every natural number s there exists a natural number ms such
that the equation X (n) = ms has more than s solutions in natural num-
numbers ii...
Thus we conclude that q
— 21cp + l, where Tc is a natural number, and so we see that each divisor of the num-
number d is of the form 2&J>+1...
+ a+1 = p (mod g); so q\p, wbich, in view of the fact that p and # are
primes, would imply q = p, and so a? = 1 (modp), that is, 2*" =
But, in virtue of theorem 6ft of Chapter V, we have 2" = 2(modj)), whence...
But, in virtue
of m\a— b, we have т|ай~- b^, which, by the formula m\ac—a? gives m\ac— 6й...
Therefore tbe numbeT of №th power residues for a given
modulus p is understood as the number of mutually non-congruent (modp)
nth power residues for the modulus p...
Consider an exponential congruence
a1 = Ь(топ.р),
where a, b are integers not divisible by the prime p...
Thus the congruence turns into the congruence
ox = 11 (mod 12), which is satisfied only for я = 7, provided the as's
are taken out of the sequence 0,1, ...,11...
In virtue of A) and B), we also have
whence mlogg' < logJV" < (•m4-l)log(? and therefore
logs
:«+i,
which proves
F)
Formulae F) and E) show that if Ж is represented as A) and condi-
conditions B) are satisfied, then the numbers иг- and cn (» = 0,1,..., m)
are uniquely defined by number N...
Therefore, by A2), we obtain
the following expansion of number ж into an infinite series:
A3)
where, by A1), numbers о„ are digits in the scale of g...
In order to рготе the
converse it is sufficient to show that if a sequence of digits сис^,.....
Clearly, the number of classes of sequences consisting of m terms is
not greater than 10*1...
For example number
1234567890
9999999999
is normal in the scale of 10; number ^ is normal in the scale of 2 but
it is not normal in the scale of 3...
Suppose that a rational number rjs, r being an integer
and, s a natural number, provides an approximation of an irrational
number m better than the n-th convergent B,n (w > 1) of x...
It can Ъе proved
that if the nimber s of the terms of the period is even, then number
Is is equal to the first index Ъ for which bk+1 = bk; if в is odd, ihen l(s — 1)
is the first index Ъ for which ek+1 = ck p)...