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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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3x2 + 5y2 + lz2 + 9xy + llyz + I3zx = 0
has a nontrivial integral solution...
If / is of degree 1 we call €f a line, if / is of
degree 2 we call €f a conic or quadratic...
In the latter case every point of the line is also on the
curve iff(M), and we say that L is a component of ^(R)...
If all points of a
curve are simple, including any points at infinity that may lie on the curve,
then the curve is called nonsingular or smooth...
Let f(x, y) be a quadratic polynomial
with rational coefficients, and suppose that the curve ^(R) contains a
rational point (x0, y0)...
The
problem of finding the singularities of a curve is also easier in projective
coordinates...
If Zo Ф 0, then the affine curve
i?f(Q has a singularity at (Xo/Zo, Yo/Zo); but if Zo = 0, then €f(Q has a
singularity at the point at infinity Xo : Yo : 0...
Proof Since L is not a subset of ^-(R), we see by Theorem 5.15 that L
and ^(R) can have at most three distinct points in common...
Instead, its third intersec-
intersection with the curve is at the point 1: - 1:0 at infinity...
But the
roots of a quadratic polynomial with rational coefficients are rational if
and only if the discriminant of the polynomial is the square of a rational
272 Some Diophantine Equations
number...
We now derive explicit formulae for the coefficients of P! + P2 on an
elliptic curve, in terms of the coefficients of Pj and P2 and the defining
equation of the curve...
The Lutz-Nagell Theorem Let -tff(U) be an elliptic curve given by an
equation of the form E.50) with integral coefficients...
These groups are of various orders, and we expect that some of
them will yield a factorization of m very quickly...
If fix, y) is a polynomial with integral coefficients, of degree 3 (mod p),
if fix, y) is irreducible (mod p), and if the projective curve €fiZp) has no
singular point, then we call this curve an elliptic curve modulo p...
Show, conversely, that if и is an integer such that u2 + 1
# O(modp), and if v, x, у are given in terms of и as above, then
x2 + y2 = l(modp) and x # l(modp)...
In our discussion following the proof
of Theorem 5.24, we found that this quartic is birationally equivalent to
the elliptic curve у2 = хъ — Ax...
Thus Catalan's question is resolved, apart from a certain finite
calculation, which, however, is too long to perform...
Kummer
developed the arithmetic of integers in algebraic number fields and formu-
formulated a criterion, which if satisfied, guarantees that Fermat's equation has
no solution...
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...
This area traces its roots to a time just a century ago when the
properties of Q-birational equivalence were first investigated by Hilbert,
Hurwitz, and Poincare...
All the fractions that appear are in reduced form; all reduced
fractions a/b such that 0 < a/b < 1 and b < n appear in the nth row; if
a/b and a'/b' are consecutive fractions in the nth row, then a'b — ab' = 1
and b + b' > n...
Solving, we find x = a + а', у = b + b', and hence
(a + a')/(b + b') is the unique rational fraction lying between a/b and
a'/b' with denominator b + b'...
In view of what was just proved about the
6.3 Irrational Numbers 309
cosine function, we need look only at the possibilities cos20 = O,
±1/2, ± 1...
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