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Sierpinski W. Elementary theory of numbers (Warszawa, 1964)(L)(T)(224s).djvu |
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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
I
22fi
СНДРТЕК V...
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
d\n
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...
As
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:
A3)
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...
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...
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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