Diophantine Geometry

Integer solutions on algebraic curves — lattice points and geometry

12
25
Integer solutions appear here
x² + y² = 25 — Pythagorean triples / sums of two squares

Diophantine equations seek integer (or rational) solutions to polynomial equations. The circle x²+y²=n has integer solutions iff every prime p≡3 (mod 4) appears to an even power in n (Fermat's two-square theorem). Elliptic curves y²=x³+ax+b are the frontier of number theory — integer points are finite (Siegel's theorem) except degenerate cases. Pell's equation x²-Dy²=1 has infinitely many solutions via continued fractions. Hover over the curve to see nearby lattice points.