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Niven I., Zuckerman H.S., Montgomery H.L. An introduction to the theory of numbers (5ed., Wiley, 1991)(K)(T)(541s)_MT_.djvu |
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Size 3.4Mb Date Jan 8, 2003 |
curve iff(M), and we say that L is a component of ^(R)...
In view of the definition of M, if n < M
then all coefficients of qn(u) are 0, and hence qn(m) = 0 for all m...
Since
/A, у) is of degree 2 at most, there may exist one or two rational values of
m, say mx and m2, for which /A, m) = 0...
Thus we
assume that €f contains no line, so that L intersects €f at a unique third
point of €f, which we denote AB...
It is not hard to show that if
such a polynomial exists, then its coefficients lie in the same field as the
field containing the coefficients of / and g...
Let fix, y) be a polynomial of degree d with real coefficients, and set
p(t) =filt/i\ + t2\(X - t2)/(l + t2)Xl + t2Y...
Starting from the triple B, — 1,1),
show that this generates infinitely many distinct rational points on the
curve x3 + y3 = 7...
Theorem 5.17 Let fix, y) be a cubic polynomial with complex coefficients
(which may in fact all be rational or real)...
From the first of these relations we deduce that Zo =
+ i/3X0, and from the second we see that Zo = + i/3Y0, so we deduce
that Yo = ±X0...
Proof Let L denote the line passing through A and 0, which therefore
contains the third point АО, as in Figure 5.4(a)...
Thus we see
that each of these four points can be written in precisely one way in the
form m\ + nB with m = 0 or 1, n = 0 or 1, and the group Ef(Q) is
isomorphic to C2 Ф C2...
By means of this correspondence, we discover that
Theorem 5.10 and Theorem 5.24 are equivalent...
Let Pl = (д:,, yx) and P2 = (x2, y2) be two points
on this curve, and put P3 = Px + P2 = (x3, y3)...
Proof Let Z be the least common denominator of the rational numbers
xvyv so that xx=X/Z, yx = Y/Z with Z > 0, g.c.d.^Y, Z) = 1...
The
elements g, are not uniquely determined, but in any such presentation of
the group the number r is the same...
Working modulo m, we
calculate a highly composite multiple of some initial element, in order to
find the identity in the group...
Condition E.60) fails if AA s
27 (mod p), but since we have already determined that m has no prime
divisor less than 10000, it follows that g.c.d...
Thus Catalan's question is resolved, apart from a certain finite
calculation, which, however, is too long to perform...
Moreover, it may be shown that if
A = (p(t), p'(t)) and В = (p(u), p'(u)\ then A + В is given by
(pit + u), p'U + u))...
One way to complete the proof of the associativity of addition of
points on an elliptic curve (mod p) involves observing that the field Zp is
contained in its algebraic closure Tp, which is an infinite field...
On the other hand,
replacing x by a/b in F.6) and multiplying by b"/a, we observe that
cob"/a is an integer, so a\c0...
We apply Lemma 6.17 with g(x) in the
form (a — bx)" to conclude that x"(a — bx)" and all its derivatives,
evaluated at x = 0, are integers divisible by и!...
Theorem 6.25 Let xv x2,- ■ ■ ,xk be arbitrary real numbers, and let n be a
positive integer...
The only representation of 2 as a
sum of four squares subject to F.12) is 2 = I2 + I2 + 02 + 02...
Furthermore, a, > 1 for i > 1 because a,_1 = [f,--i] and the fact that
6_i is irrational implies that
0
6=
Next we use repeated application of G.7) in the form £, = a; + l/6+i
to get the chain
1
^ = ^o = flo+ 7~ = <flo>£i>
bl
0.fll + 7~
62
This suggests, but does not establish, that £ is the value of the infinite
continued fraction (a0, ax, a2, • • • > determined by the integers a,...
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