The Simplest Rational Right Triangle with Area 157

The Congruent Number Problem [Up]

Congruent Numbers

For any positive integer n, let E_n denote the elliptic curve defined by the equation y² = x³ - n²x. Then n is a congruent number if and only if E_n has infinitely many solutions. The curves E_n are quadratic twists of the elliptic curve y² = x³ - x, which is isogenous to the modular curve X₀(32).

Links

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• Is n congruent? Enter a number in the box and hit return to find out if it is a congruent number (the Birch and Swinnerton-Dyer conjecture is assumed):    (this not written yet)
  
  
  
• Neil Koblitz wrote Introduction to Elliptic Curves and Modular Forms, Springer, New York, 1984.
  
  
• Notes from a talk that William Stein gave on congruent numbers:
  • congruent_number.ps (postscript file),
  • congruent_numbers.sdd (StarOffice presentation).