| A modern reason for being interested, wildly over-simplifying[0] ... Consider an equation of the form y^2=ax^3+bx+c, and consider the points (x,y) where x and y are rational. There may be none, there may be finitely many, there may be infinitely many. Take a huge, structureless[1] prime p. Any rational r/s can be thought of as r times s^{-1} modulo p, so rationals are roughly the same as integers when you work modulo a prime. So the rational solutions to the equation above (which, by the way, is an elliptic curve) give us integer solutions when we work modulo p. And now by using the geometry of the curve we get a group where the elements are pairs of integers. That's because we found rational solutions. Suddenly everywhere we use groups - such as in cryptography - we can use these numbers that have arisen as rational solutions to an equation. So being able to find rational solutions to equations is useful. [0] With any luck someone more knowledgeable can fix the worst of the errors in this. [1] So not of any particular form, such as 3^k+1 or similar |