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Sierpinski W. Elementary theory of numbers (Warszawa, 1964)(L)(T)(224s).djvu |
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every integer x we have
ж" = х{тяй.р), да" =ajs(mody), and so on...
We as-
assume that a;^=0(modm), since otherwise if a = 0(modm), D3) be-
becomes a congruence of degree less than two...
It follows immediately from the definition of q>(n) that y(l) = 1,
<р{2)=1, ф) = 2, срЩ=2, yE)=4, yF)=2, <p(l) =6, v{8) = 4,
?(9) = 6, pA0) = 4...
If ян, m%, ..., щс are natural numbers any two of which
are relatively prime, then
low let та be a natural number >1 and n = !Й1
Й'' its fac-
factorization into prime factors...
On the other hand, it can be proved
that there exist infinitely many even natural numbers m for which the
equation cp (x) —m has no solutions in natural numbers x...
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)...
Consequently, the number of the remainders
is equal to the number of the numbers g°, gl, <f,..., gp~*, i.e...
Thus we see that for any integer a the relations 6|a(a*2— 1) and
7(a(oc—1) simultaneously hold, which, by {6, 7) = 1, gives 42]a(a4: — 1), and this
proves that number 42 has property P...
Consequently number 2 belongs to an exponent < p— 1
with respect to the modulus p and is not a primitive root of p...
But, by theorem 11 (with d =
[%,p — 1) =2), there are only \[p—-1) quadratic residues in the se-
sequence 1,2, ...,p — 1...
Од the other hand, by the tables pre-
seuted above, we see that ind3 =4, whence, putting inda; = 2/, we
obtain the congruence %y = 4 (mod 12)...
Prove that the last digit of the representation as a decimal in the scale of
12 of any arbitrary square is a square...
In one of
them all cu's except a finite number are equal to zero, in the other from
a certain n onwards all cm's are equal to g—X...
The first effective example of an absolutely normal number was given
by me in the year 1916 (Sierpinski [5], see also H...
we find the least
natural number &a such that 1сгХх > 1 and we put TcsWx =l+x% and so
on...
Equality E) remains valid
if am is replaced by «„,4 — on each side of the equality (since aro+1 > 0)...
Let x be a given irrational number that is represented as a con-
continued fraction as in A2), and let rjs be a rational number that approxi-
approximates x better than the nth convergent Д, of a...
Other quadratic irrationals do not have these properties; for example,
2+1/19
= @; 1,6,1,1,1),
= @; 1,3,1,2,8,2)...
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...
Continued fractions of quadratic irrationals
303
sisting of n + 1 terms, each of the first и terms being equal to 1 (cf...
But the defi-
definition of p shows that p is not of the form 87c +1, and so it must be of the
form 87c — 1...
/Z>\
First of all we note that the task of calculating the value of l-^-l,
where P is an odd number > 1 and D an integer relatively prime to P,
may be reduced to that of calculating the value of l-^)i where Q is an
odd natural number...
Therefore, in order to apply theorem 3 while investigating whether
a given number Mp (p being a prime > 2) is a prime or not, we proceed
as follows...
For example, the prime divisors of number Ft are, by
theorem 6, of the form 26fc+l = 64Ж+1...
For any of the
numbers n = 5, 23, 73, 125, 1946, wo have б-2и+2 + 1 [ Fn and also
!>-23'+l | Fie...
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