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## Niven я., Zuckerman H.S., Монтгомерай Х.л. Введение в теорию чисел (5ed., Wiley, 1991) МП

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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Cites: By considering the intersections of a line with the given curve €f, we
may hope to generate new rational points on €f from those already
known...
We note, moreover,
that the coefficients of the fix) are determined by m, r, and the coeffi-
coefficients of fix, y), using only multiplication and addition...
E.45)
In general, the roots may be complex, but we see from this identity that if
the coefficients of p(x) are rational, and if all but one of the roots is
rational, then the last root must also be rational...
The binary operation AB on an elliptic curve does not define a group
law, because there is no point 0 of the curve with the property that АО = А
for all A on the curve...
Not only that, but if ax + by +
с = 0 defines the line L, then there is a quadratic polynomial q(x, y) such
that h(x, y) = (ax + by + c)q(x, y)...
Again, by choosing such an A' sufficiently close to A, we ensure that no
new coincidences are introduced among the P,-...
If the coefficients of / are rational, and if ^(Q) is
nonempty, then we may take 0 to be a point of ^(Q), and thus the set
€f(Q) forms a group Ef(Q) that is likewise uniquely determined up to
isomorphism...
By elementary algebra we
may verify that
-4xg{y/Bx),y2/BxJ -x/2) =f(x,y),
and that
fBu2 - 2v,4u3 ~ 4иу) = -8(u2 - v)g(u,v)...
We assume for the mo-
moment that xx Ф x2- Let m denote the slope of the line through these
points, m = (y2 — У\)/(х2 — Xy)...
(For example, G might be the group tors(£^(Q)).)
Show that if ifAU) has one connected component then G is cyclic,
and that if tff(u) has two connected components then either G is
cyclic or is isomorphic to Cn/2 Ф C2...
Thus a line (mod p)
is the collection of pairs (x, y) satisfying a congruence ax + by + с =
0(mod p), where p does not divide both a and b...
When using these
formulae to factor a number m, we proceed with congruences modulo m,
and use E.55) if uxv\ # u2v2(mod m)...
A double point (x0, y0) of ^(C) is called ordinary if the quadratic
form
a2/ df d2f
() + 2() +
has distinct roots...
That is, the only integral solutions of the equation
xm _yn = l with x > 0; у > o; m > l, n > l is 32 - 23 = 1...
Kummer called these numbers "ideal elements," but it was later found
that the same effect can be achieved by manipulating certain sets of
numbers within the original algebraic number field...
An indefinite elliptic integral is not an elementary function, but it can be
represented by introducing a new transcendental function, the Weierstrass
p-function, which satisfies the differential equation p'2 = p3 — Ap — B...
Since most machines perform multiplication much
more slowly than addition, a rough measure of the time required to
evaluate a typical expression is obtained by simply counting the number of
multiplications involved...
In its full
strength, it is not restricted to plane curves, and it applies to points whose
coordinates lie in any fixed algebraic number field...
Proof Replacing x by a/b in F.6) and multiplying by b"~l, we note that
cna"/b is an integer, and hence b\cn since (a, b) = 1...
But — 1 < cos 0 < 1, so the only possible values of 2 cos 0, apart
from 0, are +1 and ±2...
The (/ + n)th deriva-
derivative is c{(/ + и)!}, and since j > 0, this is divisible by n\
Theorem 6.18 тг is irrational...
Show that if ■€ is convex, unbounded, and contains an interior point,
then v(i?) = +oo...
§6.4 The geometry of numbers was initiated and named by Hermann
Minkowski A864-1909), who published a book on the subject in 1894...

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