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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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Date Jan 16, 2005

Cites: Thus we arrive at the conclusion that Ър \ a3"—1
holds for any a with (a,3p) =1, and this means that 3p is а В
number...
We now prove that under the conditions of theorem IB congruence
D7) has precisely two roots...
This follows immediately from the factoriz-
factorization of the following numbers into prime factors: 5186 =2-2593,
5187 =3-7-13-19, 5188 = 22-1297 and 2592 = 2-6-13-18 = 2-1296.)
We do not know whether there exist infinitely many natural num-
numbers n for which <p(n) =<р(и+1)...
Hence ж4 = pl1+1pl2+I...¦
a-.-Pe and consequently
Hence, looking at the formula for pt—l and recalling the definition of
m, we see that <p{щ) — mfor t = l, 2,...,«...
Prove that for any natural number it there exists at least one natural num-
number n such that cp(n-j-k) = g? (n)...
It is easy to prove that if n — g?1 ??••¦?? w tbe factorization of
the number n into prime factors, then
We also have
d\n
g prime
§ 3...
If q = 3, then 2-1 =¦ 0(mod9), whence 22p = I(mod9) and, by theorem 9,
I
248
СЯАРТЕВ VI...
Then, in virtue of the theorem of Format, p\cS>-1— 1, _p jb|"—x— 1, whence
pla®-1— Щ~1...
Fw any natural number n th?re exist infinitely many
prime numbers of the form nfc + 1, where к is a nai/nral number...
For each natural divisor д of number p — 1 denote by f(8)
the number of those elements of sequence B3) which belong to exponent
6 with respect to modulus p...
But jJ— 1 = 3 is
impossible because p2 is a prime, so p2 — ^ ~ ® ^ valid, whence p2 ~ 7 and conse-
consequently m = 2- 3 ¦ 7 = 42...
IE each number of the sequence
0, 1, 2, ..., </-l
C)
is denoted by a special symbol, the symbols are called the digits
formula A) can be rewritten in the form
and
There yn is the digit which denotes the number cn...
We have tie following sequence of
equalities:
Ж = с„ + дЖ1, Ж1=о1+дЖ2, ..., Жт_1=ст_1+дЖт, Жш = ст...
For example, if n = 3 we have the sequence 3, 9, 81, 65, 61, 37,
58, ..., 16, 37, ...; if n — 5, we have the sequence 5, 25, 29, 85, 89, 145, ...,
58, 89, ...; if я = 7, we have the sequence 7, 49, 97, 130, 10, 1, 1, 1, .....
Then ж = rn-\-ljgn, and so, by
(li), ж is the quotient of an integer by a power of пшпЬет д...
A number > 4 whose digits (in
the scale of 10) are all equal to 4 is divisible by 4 but not divisible by 8...
Consequently,
for да = 1 the probability is equal to \, for m = 2 it is \; for да = 3
it is only j?, and so on...
Then the denomina-
denominator s of the rational number rjs is greater Лап the denominator of the
convergent En...
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...
Let VT> =
(a0; alf a2, ¦¦.,«<) be tbe simple continued fraction for V3, and PkjQk
the fcth convergent to it...
It follows from what we stated above that number h is equal to the num-
number of the terms of the period of the simple continued fraction for VT>...
+ Д(р_1)B is
exactly the number of the remainders obtained by dividing the numbers
of F) by p, successively...
Among the pairs 2Б7, 127; 127, 3; 3, 1, only the second is such that each of its terms
is of tlie form 4i + 3...
Hence it follows that nn+1 = Я&+а- ТЬив, for s = 0 we obtain
number Jfx = S, for s = 1 number F3 ¦= 257, for s = 2 and s = 3 numbers J?6 and
JPii, which are oompoeite; for s = 4 we obtain the number Рэд > 22 > 210, but
this has more than 3000OO digits (Sierpiuski [19])...



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