Pries: 405 Number Theory, Spring 2012. Homework 5.
Due: Friday 4/6.
Rational points and Pell’s equation
Read: Handouts.
Do 6 of the following problems.
1. Find the limit as t → ∞ of the rational point
(x, y) = (
1 − t2 2t
,
),
1 + t2 1 + t2
that we found on the circle x2 + y 2 = 1. Explain this limit geometrically.
2. Use lines through the point (1, 1) to find a formula for all rational points (x, y) on the
circle x2 + y 2 = 2 (need x and y to be fractions).
3. Use lines through the point (−1, 0) to find a formula for all rational points (x, y) on
the hyperbola x2 − y 2 = 1.
4. Find four solutions (x, y) with x, y positive integers to x2 − 5y 2 = 1.√Hint: use trial
and error to find a small solution (a, b) and then take powers of a + b 5.
5. Find the sum of the rational points (1, −3) and (−7/4, 13/8) using the group law on
the the elliptic curve y 2 = x3 + 8.
6. The point P = (3, 8) is a rational point on the elliptic curve y 2 = x3 − 43x + 166.
Compute 2P , 4P , and 8P using the elliptic curve duplication formula. Comparing 8P
with P , what can you conclude about the order of P ?
7. Graph the curves y 2 = x3 and y 2 = x3 + x2 . Explain how these graphs are different
from the graphs of elliptic curves drawn in class.
8. Show that the equation y 2 = x3 + x2 has infinitely many integer points (x, y). Hint:
try y = tx.
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