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Sierpinski W. Elementary theory of numbers (Warszawa, 1964)(L)(T)(224s).djvu |
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definition of д implies that 3 < p— 1 which gives 1 < S <p, contrary to the defi-
definition of the prime p...
The number of the natural numbers r < I for which (r,l) =1
is oi course <p(l), this being the number of the columns in which all the
I
230
CHAPTER VI...
For an arbitrary natu-
natural number a > 2 we have a—1 > \'a, and for any natural number 6
the inequality b — | > J6 bolds...
On the other hand, it can be proved
that there exist infinitely many even natural numbers m for which the
equation cp (x) —m has no solutions in natural numbers x...
(The
answer is in the positive, provided any even natural number > 6 is
the sum of two different prime numbers.)
EXERCISES...
Therefore n — 3№+2 or n
_ 2'36*+2 and, as is easy to verify, in any case ip(n) = 2-36к^ — т...
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...
of Carmichael (and so absolutely pseudo-prime) it is necessary and suf-
sufficient that X(m) \m—l (cf...
If q = 3, then 2-1 =¦ 0(mod9), whence 22p = I(mod9) and, by theorem 9,
I
248
СЯАРТЕВ VI...
Since the number of them is cp{m), this being equal to the number of the
numbers relatively prime to m which appear in the sequence 1,-2, ...,m,
then for any integer a relatively prime to m there exists precisely one
number у in the sequence 0,1,2,..., q>(m) — 1 such that g" = a;(modm)...
Considei an arbitrary infinite sequence c1; c2,..., wheie en (n — 1,2,...)
are digits in the scale of g...
The first effective example of an absolutely normal number was given
by me in the year 1916 (Sierpinski [5], see also H...
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...
It follows from B) that the ftth convergent Bk is a function of ft+1
variables, a0, %,...,%, aad that if foi ft < n number ah is replaced by
number ak-\ , the convergent Mk turns into the convergent Rk+l...
Рог irrational numbers of the 2nd degree representations as simple con-
continued fractions are known...
Let x be a given irrational number that is represented as a con-
continued fraction as in A2), and let rjs be a rational number that approxi-
approximates x better than the nth convergent Д, of a...
Therefore, in virtue of the relation xn = an+l/xn+1 and C1), we in-
infer that а%_! = ^^„i...
The number /919 has a period consisting of 62 terms:
l/919 = C0; 3,3, l72,l, 2,1,1,1,2, 3,1,1,19, 2, 3,1,1,4, 9,1,
7,1,3,6,2,11,1,1,1,29,1,1,1,11,2,6,3,1,7,1,
9, 4,1,1, 3,2,19, l,'l, 3,2,1,1, 1,2,1,2,1,8, 3,60)
(cf...
But, if * is even, then, by D4), none of the (sn — l)-th convergents
gives a solution of equation F1)...
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...
SierpinsH [2]) that if \ — t is a real number defined for
л fixed odd prime p and any integer D not divisible by p, which is different from zero
for at least one value of D and different from 1 for at least one D and which, more-
moreover, satisfies the conditions
1° if B = B'(mody), then j—J = j—J,
fBD'l fBl fB'l
2° I 1 = j—11—| for any D and V that are not divisible by p,
then for any integer V not divisible by p we have
Let g be a primitive root of the prime p...
To
\pl
see that this in fact implies the equality ( —I = (—1)я, asserted by the
\pl
lemma, it is sufficient to note that —I is equal either to 1 or to —1
\p!
and that p, being an odd prime, is > 3...
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