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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: Therefore, in either case, the relation jj]o193— a holds
for any integer a and p = 3, 5, 13...
Hence, since 29—1 = 7-73 and 229—1 = 233-1103-2089, it follows
that 73I2™—1—1, llOSja"-1— 1 and since 20891239— 1, 208912"-1— 1...
For i = 2, 3,...,»-}-1 we then have p | (аг{—xjgfa), which,
since atj, ж2,..., жй+1 are different roots of congruence C7), implies that
p\g{xi) for i = 2, 3,..., w-j-l...
Since В
is odd, number z must also be odd, whence, since the square of an odd
Elementary theory of numbers 15
The number of the roots of the congruence is equal to 2д+;1, where I is
the number of odd prime factors of the number m and p = 0 for m not divis-
divisible by 4, ft = 1 for m divisible by 4 but not divisible by 8, and, finally, /л = 2
for m divisible by 8...
Suppose that »>1 and let » = й^й^-'-й** be
the factorization of number n into prime factors, a0 being a non-negative
integer and a^, at,..., а„ natural numbers...
Therefore /3=1, whence
p =2-53i:+l which is impossible since the number 5~k = (a*J is con-
congruent to 1 with respect to the modulus 3, whence 3 | p, so p =3 and
this is clearly false...
It is easy to prove that if n — g?1 ??••¦?? w tbe factorization of
the number n into prime factors, then
We also have
g prime
§ 3...
So tit2-..tB sa a3 i
1 (modr), whence,
in view of (a, r) = 1, by theorem 8, we infer that aPW ™ l(modr)...
Thus we see that our assumption leads us to the
conclusion that there exists a solution r of congruence A9) less than d,
which contradicts the definition of 8...
it is easy to see, the numerator divided by b1 leaves a remainder equal
either to 6—1 or to 62 — 6+1...
;p — X j (Л—ZI*, which, in virtue of the relations n = dm, p — l = ds,
gives s | Gc — Z)m; so, since (to,s)=1, s | 7c—Z, which is impossible
because 7c and I are two different numbers of the sequence 1,2,...,«...
Consider an exponential congruence
a1 = Ь(топ.р),
where a, b are integers not divisible by the prime p...
These are: periods consisting of
one number, which can be any of the numbers 1, 1634, 8208, 9474; a period con-
consisting of the numbers 2178, 6514; a period consisting of seven numbers 13139, 6725,
4338, 4514, 1138, 4179, 9219 (see also Chikawa, Iseki, Kusakabe and Shiba-
mnra [1])...
Therefore, by A2), we obtain
the following expansion of number ж into an infinite series:
where, by A1), numbers о„ are digits in the scale of g...
If 44...4 = ma, then
the number 111...1 would be a square,- but this is impossible since the last two digits
of a square oi a natural number cannot be 11...
This, combined "with inequality A5), indicates that
the even convergent» increase strictly as they tend to x, while the odd
convergents decrease strictly...
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...
Since, in view of theorem 3, the sequence %, aa must be symmetric, we
have ax = a2 and, moreover, аг Ф 2a0 since otherwise the period of the
simple continued fraction for YT> would consist of one term at...
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...
?or eiample, the fraction j-' -\ ¦ + -^ has the value 2, but the con-
convergent t— -\ has no value...

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