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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: The notation <p{n), however, is due to Gauss (it was
introduced by him in 1801) — this is the reason why some authors call
the fimction <p(n) Gauss's function...
But the number of numbers B) is m,
and this is equal to the number of the residues modm, i.e...
The equation cp(n) = <jj(»+1) in natural numbers n has been a sub-
subject of interest for several authors (cf...
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...
If there existed an odd natural
number да such that <p[n) = w, then n would be the product of different odd prime
factors which, in addition, would be of the form Жк = 227>+1...
In virtue of exercise 3, the number щ is a divisor of a number
m whose digits (in the scale of ten) are equal to 1...
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)...
Hence, using tbe relations a = ga = (^(mo&p), we find
а=адк№~1'>''=ды+к<-р-11''=;д'а"'=(ды)п(шо6.р), which proves that a
is an rath power residue foT the prime p...
Consider an exponential congruence
a1 = Ь(топ.р),
where a, b are integers not divisible by the prime p...
Any natural number may be uniquely expressed as a deci-
decimal in the scale of g (g being a natural number > 1), i.e...
These are: one term
periods, the term being any of the numbers 1, 153, 370, 371, 407; period consisting
of two numbers, either of 136 and 244 or of 919 and 1459; finally, periods consist-
consisting of three numbers, either of 65, 250, 133 or of 160, 217, 252 (see also Iseki [2])...
Representation of numbers by decimals
But, in view of A4), rn~rn_l = -Ц for any n — 1, 2, ..., whence c,,
a
»Л-Я*"!гы which, by A5), implies
A6)
<¦„ =
This shows that any real number ж which is not the quotient of an inte-
integer by a power of g has precisely one representation as series A3), where
<л are integers satisfying conditions A1)...
The expansion thus obtained for number e is as follows:
111 1
11-1 1-1-2 1-1-2-3
Let a be a natural number > 2...
Therefore formula B5) gives us an expansion of the irra-
n->oo
tiona.1 number x0 into an infinite Tapidly convergent series
B6)
r - X 1+1 X 4-
x0 = ¦ 1 1- • • •
% a a a
where а„ (m = l,2,...) are natural numbers satisfying the inequali-
inequalities
B7)
for и = 1,2,.....
JPk and Qk being defined by C), for an arbitrary а„ and positive
%, a2, ¦..,an те have
ft
§ 2...
The greatest of them is the
number % = 20776; all natural numbers <34 appear among the <j4's
and number 1 appears 393 times...
Then the denomina-
denominator s of the rational number rjs is greater Лап the denominator of the
convergent En...
On the other hand, if for some natural numbers a0 and at Ф 2a0
number D of D1) is natural, then, since 2a0a1+l is odd, number a\+l
(as a divisor of it) must also be odd; so number ax is even and, since num-
number D of D1) is an integer and, consequently, —\-^ 1 =
——^— is an integer, number «i + l divides number (aa—a1j2Ja1...
In view of formulae C) we have
Hence
«n-I «П-!
But, since the sequence «цЛ,,...,^ is symmetric, this gives
\an
whence Pn = Qm_!, which -was to be proved...
If m is a real irrational number satisfying the equation Aa?-\-Bso-\-C
= 0, where A,B,G are integers, then, as is known, D — Б2 — 4Л0 > 0
and Dis not the square of a natural number...