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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: We as-
assume that a;^=0(modm), since otherwise if a = 0(modm), D3) be-
becomes a congruence of degree less than two...
But the
square of an integer is congruent with respect to the modulus 4 either
to- zero or to 1...
Then, dividing tbe mimbers
B) r, l+r, 2l + r, ..., {m—l
by m, we obtain the set of remainders
C)
0, 1, 2, ..., m...
It follows from F) that if <p(n) = 2-51*, where
h is a natural number, then n must have precisely one odd prime divisor...
If i = 4, then h + h + h — 10, which is impossible because j5j, ?2, /Sa are dif-
different numbers chosen out of the sequence 1,2, 4, 8...
Formulae (8) and C7)
of Chapter IT give together the formula
(9)
cp[n) =
d\n
valid for all natural numbers n...
On the other hand, it is clear that congruence A9) has natu-
natural solutions only in the case where (a,m) =1...
It is also plain that without loss of generality we may suppose that
n > 2 (for in the sequence of odd numbers there exist (as we know) infi-
infinitely many primes)...
Every prime number p has cp (p — 1) primitive roots among
the terms of the sequence 1,2, ...,p — 1...
That is that
m = ZpifvPs, where 2 < Pi < Pz <2V Then, as in the above argument, we infer
that px — 112...
Then theTe exists an inte-
integer x, of course not divisible by p, such that a = s"(modj>)...
Prove that a necessary and sufficient condition for an integer g relatively
prime to ail odd prime p to be a primitive root of p is the validity of the relation
^(P-1}/^^ l(mod_p) for auy prime divisor q of the number p—1...
These are all the known primes of
this form, we do not know whether there exist any other such primes...
In virtue of what we
have proved above, different expansions of the form (*) give different n'a...
Then ж = rn-\-ljgn, and so, by
(li), ж is the quotient of an integer by a power of пшпЬет д...
As an immediate consequence of theorem 3* we note that if a number
as has a non-periodic representation as a decimal in a scale of g, then x is
irrational...
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...
It can be proved that the class of positive irrational numbers which
have these properties coincides with the class of the square roots of ration-
als greater than 1...
To see that sucb a representation exists we note tbat, if Tc—1
were even, then for 6t_! > 1 the number S1+
-1+ pr could be written in
place of Ък_и and for Ък_г = 1 the number й&_2 + 1 could be written in
place of &fc_j+ ...
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...
Since j»-1 = l(modji), Ъу 1° and 2°, the equalities a"-1 = j—J
= i- 1 = j —| hold, but, in view of 2°, |-i| = |—J, whence I — \ = 0 or -J—| = 1...
But, since for rk>pj2 we have Д* = 1, the formula (— 1)J*
Since the formula proved above holds for any Ъ = 1, 2,.,., (p —1)/2,
we have
(—II = (-i)h
Thus the lemma of Gauss implies
(p-ll/2
.2
Cobollaby...
Hence
¦PI
By A3), these two formulae show that the formula
is valid for any two different odd primes p and q...
The primes Жмт, Мт and Mnm which are now the largest
known primes were discovered by using the Iujiao II at the Digital
Computer, Laboratory of the University of Illinois...



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