What is the congruent problem?

Plane Geometry Read more from
Chapter Geometry and Trigonometry

It may not be something that kept you up at night, but the congruent problem has finally been solved—at least the first one trillion cases. Simply stated, the problem concerns the areas of right-angled triangles to determine which whole numbers can be the area of a right-angled triangle whose sides are whole numbers or fractions. In fact, the area of such a triangle is simply called a congruent number.” For instance, the area of a 3-4-5 right triangle is 0.5 × 3 × 4 = 6, with 6 being the congruent number.

The problem is an ancient one, and like many “Holy Grails” of mathematics, this problem was first stated centuries ago by astute mathematicians—this time, Persian mathematician al-Karaji (for more about al-Karaji, see “History of Mathematics”). In 1225, Fibonacci (for more about Fibonacci, see “History of Mathematics” and “Mathematics throughout History”) also tried to work on the problem. Many other mathematicians followed, but it took until around 2009 for an international team of mathematicians, using state-of-the-art computer techniques, to find the first trillion congruent numbers.

In order, the first few congruent numbers known are 5, 6, 7, 13, 14, 15, 20, 21, and so on. But, as one can imagine, there are so many more such numbers. The researchers were able to figure out a computer compilation method that would allow them to not only find but also verify more congruent numbers. They found 3,148,379,694 congruent numbers up to a trillion. The mathematicians were no doubt grateful for the computers, too. According to some researchers, the numbers involved are so huge that if they were written by hand, the numbers would stretch to the Moon and back.


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