#number-theory

#number-theory

Five years ago, mathematicians Dawei Chen and Quentin Gendron were trying to untangle a difficult area of algebraic geometry involving differentials, elements of calculus used to measure distance along curved surfaces. While working on one theorem, they ran into an unexpected roadblock: Their argument depended on a strange formula from number theory, but they were unable to solve or justify it. In the end, Chen and Gendron wrote a paper presenting their idea as a conjecture, rather than a theorem.

In retrospect, he's glad. "Part of my luck was that I couldn't keep up with them," he said. "They were proving theorems, but they had nothing to do with the essence of the situation." Hofstadter instead decided to test out a more down-to-earth approach. Rather than proving theorems, he was going to crunch some numbers using an HP 9820A desk calculator-a computerlike machine that weighed nearly 40 pounds and could be programmed to perform complex computations.

One of the first theorems anyone learns in mathematics is the Pythagorean Theorem: if you have a right triangle, then the square of the longest side (the hypotenuse) will always equal the sums of the squares of the other two sides. The first integer combination that this works for is a triangle with sides 3, 4, and 5: 3 + 4 = 5.