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Monday, July 01, 2019

A Thousand Years of Congruent Numbers | Roots of Unity - Scientific American

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Evelyn Lamb, Freelance math and science writer based in Salt Lake City, Utah explains, These integers have inspired one of the most important unsolved problems in mathematics.

Congruent numbers are all about right triangles, and so is this parquet floor!
Photo: Evelyn Lamb

On our most recent episode of My Favorite Theorem, my cohost Kevin Knudson and I talked with University of Montreal math professor Matilde Lalín about her favorite bit of math, the congruent number problem. (You can listen to the episode or read a transcript at

A congruent number is a positive whole number that can be the area of a right triangle with rational side lengths. So the number 6, for example, is a congruent number because it is the area of a 3-4-5 right triangle. The congruent number problem is to figure out what positive whole numbers are congruent numbers.

When I heard the term congruent number, my first question was, “Congruent to what?” But even though there are triangles involved, the name doesn’t come from the idea of two figures being congruent but from the Latin word congruum. Fibonacci used the term, which means something like “agreeing” or “harmonious,” to refer to the differences of squares in arithmetic progressions...

As Dr. Lalín told us, mathematicians have been trying to find criteria for determining whether a given number is a congruent number for a long time. Some facts are known: no squares are congruent numbers; if a prime number is 3 more than a multiple of 8, it is not congruent, but its double is; every congruent number is a congruum multiplied by a square of a rational number. That last one might seem like a jackpot, but there are a lot of squares of rational numbers, and it’s not easy to see whether a given rational number times a congruum will yield a congruent number.

Source: Scientific American