Nathan McNew, Ph.D., of 撸先生鈥檚 Department of Mathematics, and local high school graduate Jai Setty have made a breakthrough on a problem mathematicians have been working on since the 1930s, and it is all about a single, precise number that describes how common "abundant numbers" are.

Take the number 12. Its proper divisors, the numbers that divide evenly into it, are 1, 2, 3, 4, and 6. Add those up and you get 16, which is more than 12 itself. That makes 12 an "abundant" number. Abundant numbers are part of a family of curiosities that have fascinated mathematicians since ancient Greek mathematicians investigated "perfect numbers" the numbers that equal exactly the sum of their divisors like 6 (1 + 2 + 3 = 6, making 6 a perfect number). Abundant numbers are their more common cousins, where the divisors add up to more than the number itself. Since the 1930s, mathematicians have wanted to know exactly how often abundant numbers appear as you count higher and higher.

That question turns out to have a fixed answer, a numerical constant, like pi, that never changes. The search for its precise value began in 1932, when the mathematician Felix Behrend established the first rough bounds: the constant was somewhere between 0.241 and 0.314. Over the following decades, successive researchers narrowed the range. By the 1990s mathematicians knew the constant starts out with the digits 0.247... Then in 2010, Mits Kobayashi got one more digit 0.2476... Each improvement required both new mathematical ideas and the computational power to carry them out.

McNew and Setty didn't just add one more digit. Starting from where Kobayashi left off, they pushed the known value to 0.2476196..., a jump of three digits at once, and the largest single leap in precision the field has seen. To do it, they developed new computational methods future mathematicians can continue to build on.

The constant tells us that abundant numbers never become rare, no matter how high you count. Unlike prime numbers, which thin out as numbers grow larger, abundant numbers maintain a steady, predictable presence going on to infinity. That kind of certainty gives mathematicians sharper tools for studying the whole family of numbers that abundant numbers belong to, perfect numbers included.

Along the way, McNew and Setty also found that a related class of numbers, called primitive covering numbers, is far rarer than mathematicians had previously known, a finding that opens new questions of its own.

Setty began the project as a high school student and is now an undergraduate student at New York University (NYU). Authored by both McNew and Setty, has been accepted for publication in , a prestigious journal in computational mathematics.