One Trillion Congruent Numbers
Do you know what congruent numbers are? They are integers which are the areas of rational right triangles. A rational right triangle is a right triangle for which none of the 3 sides are irrational numbers.
You’re already familiar with some rational right triangles. The first one you learned about was probably the famous 3,4,5 triangle.
The area of this triangle is 3*4/2 = 6, so 6 is a congruent number. There is actually a smaller one. The number 5 is a congruent number because of the triangle 1/2,3,4. There are obviously an infinite number of congruent numbers, but predicting which numbers are congruent is an interesting problem.
The congruent numbers form a pretty odd pattern. If you go to one of my favorite websites, you’ll find that the sequence of congruent numbers starts like this:
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, … (id:A003273)
Mathematicians have been studying this sequence since al-Karajī (ca. 1000), although he was obviously influenced by Diophantus (ca. 200).
Recently, an international team has been attacking the problem with computers. They’ve now computed the first one trillion values and written an interesting paper about their work. It gets into elliptic curves, FFTs, the Chinese remainder theorem, and lots of other cool stuff.
From a physorg article.
