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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: (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)...
Thus we
see that each root of congruence D7) is congruent with respect to the
modulus p" either to я or to —г...
This shows that in each of tbe rp{l) columns, the
terms of which are relatively prime to Z, there are <p(m) numbers rela-
relatively prime to m...
For every natural number s there exists a natural number
m such that the equation <p(n) = m lias more Hum s different solutions in
natural numbers я...
Let с denote a natural number prime
to ab (there are of course infinitely many such, numbers; in particular all the numbers
kab-T-1, where h = 1, 2, ..., have this property)...
The theorem of Enlei
243
the remainder obtained by dividing tbe number ark by m (h = 1,2, ......
Thus in order to рготе the theorem we may remove some terms at
the beginning and suppose a> 1...
It is clear that if two numbers are congruent with respect to modu-
modulus m, then they belong to the same exponent with respect to modulus
да; for, if а н= Ь(modm) and for some x formula A9) holds, then bx
= 1 (modm) (since, as we know, the congruence »si(modm) implies
the congruence a? ss ^(modm) for any x = 1,2, ...)•
Tkeokem 9...
Thus we have proved that if p is odd, then there exists at least one
prime number of the form 2psk+X; if p = 2, then there exists at least
one prime of the form 2sft + l...
If for a natural number k\n[d we have
p*'11 пы—1, then, by the identity
p\k, which is impossible, since к | n and (n,p) = 1...
Every prime number p has cp (p — 1) primitive roots among
the terms of the sequence 1,2, ...,p — 1...
In fact, it is sufficient to check whether
there exists a number w in tbe sequence 1,2, ...,p — 1 which satisfies
the congruence ж" = a (mody)...
that it be either one of the numbers 2, 3 or of the form Зй + 2,
where i is a natural number...
Erom this we infer that none of the numbers with
odd indices can be a quadratic residue for the prime p...
This is equivalent to the rela-
relation 12 1 8y — 4, which, in turn, is equivalent to S]2y—1, i.e...
are uni-
uniquely defined by number 3"; so the representation of 3?" in form A) is
unique...
Hence, since by A4) number о"У„ is an inte-
integer, we see that gnrn = [gV], this being also true for n = 0 provided
» is defined as [»]...
We have thus proved that anylirrational number x0, 0 < жс < 1, may
be expressed in form B6)...
JPk and Qk being defined by C), for an arbitrary а„ and positive
%, a2, ¦..,an те have
ft
§ 2...
Thus we see that
We are going to prove that the above formula is valid for any natu-
natural number », i.e...
It is not true, however, that the square roots of natural numbers
which are not squares are the only quadratic irrationals that possess
the properties listed in theorem 3...
For example, for Ь = 1 and n = 1, 2, 3, 4, 5, 6 we find for
< = 0,1,2, ..., respectively (cf...
We note that if the coatimied fraction F2) has a well-defined value,
then it may happen that some of its convergents do not have this property...
In view of theorem 4 of Chapter V, we have
A)
Consequently, the value of I— is 1 if and only if DC-1 divided by
\pl
p leaves the remainder 1...
Consequently, Я s=—-—
(mod2), and thus, by A2), we obtain property IV of Legendre's sym-
symbol:
йот this we infer that 2 is a quadratic residue to all primes p of
the form 87c ±1 and is not a quadratic residue to any prime p of the form
8&±3 (where It is an integer)...