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Sierpinski W. Elementary theory of numbers (Warszawa, 1964)(L)(T)(224s).djvu |
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« roots, and this, by induction, completes the proof of theorem 13...
But,
since /(«)== O(moda) and f(v) = 0(modb), we have /(#) = O(moda)
and f(x) = 0(modJ); consequently, since (a, b) =1 and ab = in, f(x)
— 0 (mod да)...
Therefore the total number of the numbers of the
table, which are relatively prime to m and to I, is cp(l) rp(m)...
Thus we see that the equation if (n) = 2-36t+I, where 7; is
a natural number, has precisely two solutions, n = 36J:+I and n = 2-36i+2...
The theorem of Enlei
243
the remainder obtained by dividing tbe number ark by m (h = 1,2, ......
Therefore, multiplying the last two
congruences,we obtain (ж + 20i):c+Mfc = ^(modlO) for any & = 0, 1,2, .....
Thus we see that
if (fc, 8) = 1, then rk belongs to the exponent б with respect to the modu-
modulus p...
Therefore, since ps > p2 = 7, it must he true that p3— 1 — 7,
14, 31 or 42, which, in virtue of the fact that pa is a prime, implies p3— 1 = 42...
Thus we conclude that the numbers gn, /",..., p™ divided by у yield
different remainders...
This, since n is arbitrary, shows that there exist arbitrarily large
primes of the form 6fc—1 = 3B(it—1L-1) + 3, as was to be proved...
In this way we find 2 = 2, 22 = 4, 23 = 8, 2" = 3, 2s = 6, 2s =12,
2' = 11, 2s = 9, 2" = 5, 210 = 10, 2" = 7, 213 = I(modl3)...
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...
Hence we easily obtain the desired representation, of Ж, namely Ж = co +
+ e1g+eigi + .....
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
...
Then the en's satisfy condition A1), whence
it follows that infinite series A3) is convergent and its sum x is a real
number...
The expansion of x into an infinite series thus obtained is as follows
a,= + —- + T , , +...,
where Tcn [n = 1,2, ...) are natural numbers and Лп+г > fcnforn = 1, 2,.....
Continued fractions
But, since an+i >1, by replacing n by n+1 in A1) we obtain
1
A4)
\x-S,
¦n+ll
Tbe relation §„+1 = Qaan+Qn_1 > 6» applied to A3) and A4) gives the
evaluation
(IS)
|я-Д,+1| < \x-Rn\, valid for any n = 1, 2, 3,.....
In fact, we have ax — [jKjJ = I 1; therefore in order that
Lx— a0J
% = m it is necessary and sufficient that m <l/(«— «„) < m+1, i.e...
Then the denomina-
denominator s of the rational number rjs is greater Лап the denominator of the
convergent En...
Continued fractious of quadratic irrationals 293
The representation of YD as a continued fraction is usually written
in the form Yd = («oj ai> an ¦¦¦> fflsh *be bar above the terms indicat-
indicating that they form a period...
The argument is that if % is an even natural
number and D2) holds, then a0 is a natural number, 2a0 > аг and
number D of D1) being natural...
Accordingly we assume that t and и are a solution of the equation
хг—г1>у* = 1 in natural numbers...
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...
In this connection, we present the following two identities:
From them we derive the following corollary: for any prime p of
tbe form 64 + 1 number 2p2 is a sum of three biquadrates of natural numbers...
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