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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: As a matter of fact, we show that every
number of the form n = 3p, where p is a prime > 3, is a D number...
Thus we have shown that to each pair («, »), where « is a root of
congruence D1) and v is a root of congruence D2), there corresponds a root
of congruence D0)...
Therefore the total number of the numbers of the
table, which are relatively prime to m and to I, is cp(l) rp(m)...
We conjecture that for every natural number s > 1 there exist infi-
infinitely many natural numbers m such that the equation cp (я) = то has
precisely s solutions in natural numbers n...
Hence <p{(p- Щ) = (p— l)tp(k) (this follows at once from theorem
3 — in fact, if m is a natural number such that any prime divisor of it is a divisor of
a natural number h, then tp(mh) = т/р{Ь))...
И /ij = 2, then n + j32 = 9, \$, = 4 or 8, whence a = 5 or 1 and so »
= 2s-5-17 or 2-О-257...
So, if in addition (a, m) = 1, then (am-Xm), m) = 1,
whence m|a'<1"'—1, which gives the theorem of Euler...
is an integer rela-
relatively prime to a natural number m, then the congruence
A9)
ax = l(mod»i)
lias infinitely many solutions in natural numbers x\ for example an
infinite set of solutions is formed by the numbers a: = k<f{m), where
к =1,2, .....
) the equation X[n) = 2
has precisely six solutions, n = 3, 4, 6, 8, 12, 24, and the equation
X[n) = 4 has 12 solutions (the least of which is те = 5 and the greatest
№ =340); the equation X(n) =12 has 84 solutions (the least of which
is n = 13 and the greatest n = 65520)...
But, as is easy to calculate, <5 = 6, so 6|2|>, whence Sip, and this con-
tradiets the assumption that p > 3...
Hence, in virtue of ad = l(mody), we have r| s= (as)k = l(mody), which
proves that numbers B6) are roots of congruence B5)...
Every prime number p has cp (p — 1) primitive roots among
the terms of the sequence 1,2, ...,p — 1...
We then have
q}(.p— l)/S, whence <5|(j>— l)/2 and, sinceyls'— 1 (because g belongs to the exponent S
with respect to the modulus p), then a fortiori p\ j№-4/s— 1, i.e...
This proves that for a given natu-
natural number Ж (with a fixed natural number g > 1) there is at most one
representation A) such that conditions A) are satisfied...
are uni-
uniquely defined by number 3"; so the representation of 3?" in form A) is
unique...
If,
however, m>l, then oTO_! ^ o—1, therefore, by A1), em_1 < g—X,
that is, om_i < </—2, which shows that number c'm_1 =Cm_i+l is also
a digit in the scale of g; consequently number я has a representation
...
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...
We have thus proved that anylirrational number x0, 0 < жс < 1, may
be expressed in form B6)...
It is also easy to find all natural numbers D for which the represen-
representation of Y3 as a simple continued fraction has a period consisting of
two terms...
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...
p—1 ff—1
ЖишЬег • is odd if and only if each, of the numbers p
2 2
and 2 is of the form 44 + 3; hence equality V may be expressed by saying
If two different odd primes p and q are of the form iJt + 3, then
: least one of them is of the form 4& + 1, then I —I =1 — 1...
Legendre's and Jacobi's symbols
All tbe numbers gx—1, qa — 1, ..., g,—1 are even; consequently the
product of any two of thorn is divisible by 4...