Assignment 11: Sums and differences of cubes
Paul Hewitt
8 November 2013
Every odd number is a difference of two integral squares. For example,
2k + 1 = (k + 1)2 − k 2 — but there may be many other solutions. However,
not every odd number can be written as a sum of two integral or even rational
squares. For example, there is no rational solution to the equation x2 +y 2 = 3.
Diophantos found that every number that can be written as a difference
of two rational cubes can also be written as a sum of two rational cubes. In
other words, given any (positive) rational numbers a and b with (say) a > b
then there is always a positive rational solution to the equation
x 3 + y 3 = a3 − b 3 . (1)
Diophantos used his “secant-and-tangent method”, or what we now call the
group law on the elliptic curve (1).
One solution is, of course, P = (a, −b). We want a positive solution,
but Diophantos realized that we can start with P , find the tangent line at
P , then follow it to another solution. We can repeat this process until both
coordinates are positive. Or we can choose any pair of solutions we generate
along the way and consider instead the secant line between those two.
Take a = 3 and b = 2 and apply Diophantos’ method to find positive
rational solutions to the equation x3 +y 3 = 19. Graph this cubic and illustrate
the steps in your solution.
Extra credit: Prove that there are no rational solutions to the equation
x2 + y 2 = 3. Hint: Write x = a/c and y = b/c, where a, b, c are integral,
and clear denominators in the equation. Show that if we have an integral
solution to this latter equation then we can find one where the a, b, c are
pairwise relative prime. Now consider the equation modulo 3.
