Cryptography Reference
In-Depth Information
Exercises
1.1 Use induction to show that
1
2
+2
2
+3
2
+
···
+
x
2
=
x
(
x
+ 1)(2
x
+1)
6
for all integers
x ≥
0.
(a) Show that if
x, y
are rational numbers satisfying
y
2
=
x
3
1.2
−
25
x
and
x
is a square of a rational number, then this does not imply that
x
+5and
x
5 are squares. (
Hint:
Let
x
=25
/
4.)
(b) Let
n
be an integer. Show that if
x, y
are rational numbers sat-
isfying
y
2
−
=
x
3
n
2
x
,and
x
n
, then the tangent line to
this curve at (
x, y
) intersects the curve in a point (
x
1
,y
1
) such that
x
1
,x
1
−
−
=0
,
±
n, x
1
+
n
are squares of rational numbers. (For a more
general statement, see Theorem 8.14.) This shows that the method
used in the text is guaranteed to produce a triangle of area
n
if we
can find a starting point with
x
=0
, ±n
.
1.3 Diophantus did not work with analytic geometry and certainly did not
know how to use implicit differentiation to find the slope of the tangent
line. Here is how he could find the tangent to
y
2
=
x
3
−
25
x
at the
point (
4
,
6). It appears that Diophantus regarded this simply as an
algebraic trick. Newton seems to have been the first to recognize the
connection with finding tangent lines.
−
4+
t, y
=6+
mt
. Substitute into
y
2
=
x
3
(a) Let
x
=
−
−
25
x
.This
yields a cubic equation in
t
that has
t
=0asaroot.
(b) Show that choosing
m
=23
/
12 makes
t
= 0 a double root.
(c) Find the nonzero root
t
of the cubic and use this to produce
x
=
1681
/
144 and
y
= 62279
/
1728.
1.4 Use the tangent line at (
x, y
) = (1681
/
144
,
62279
/
1728) to find another
right triangle with area 5.
1.5 Show that the change of variables
x
1
=12
x
+6,
y
1
=72
y
changes the
curve
y
1
=
x
1
−
36
x
1
to
y
2
=
x
(
x
+ 1)(2
x
+1)
/
6.
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