Professor and Student Disprove 323-Year-Old Mathematics Theorem
Graduate student Adam Krause and Associate Professor Howard Skogman developed a corrected version of a 17th-century mathematics theorem.
In his first week of classes last semester, graduate student Adam Krause was attempting to solve a problem in his number theory textbook when he noticed something
"When I initially tried to solve it, I suspected it to be false," said Krause. "So, I wrote a program and found counterexamples to it. The program that I wrote started to spit out interesting results."
The next day, Krause brought his findings to his course instructor, Associate Professor Howard Skogman, who has taught a range of mathematics courses at the College since 2001 while directing the department's graduate program.
The theorem, coined "Ozanam's Rule" by the author of the textbook used in Skogman's course, was devised by the French mathematician Jacques Ozanam (1640 – 1718). Ozanam proposed tests to determine whether a given integer was a triangular number or a pentagonal number. The tests involve number sequences that, according to Skogman, have been studied since the time of the ancient Greeks.
"Upon further experimentation with the program, Dr. Skogman and I discovered that the theorem was false in very regular ways, which suggests that there was some way to edit Ozanam's Rule to make it true," said Krause.
Krause and Skogman teamed up to investigate the discrepancies initially noticed by Krause. While Krause continued to write programs, Skogman developed theoretical approaches to solving the problem. The pair worked together throughout the semester "refining conjectures on the data" and "looking for a proof based on the evidence," according to Skogman.
"Adam and I proved that the Ozanam's test fails in general for all m at least 5," said Skogman. "We further showed that for any given m, we could predict exactly what percentage of integers that produced squares with Ozanam's test were in fact m-gonal."
By the end of the semester, Skogman and Krause had developed a revised theorem:
If m is a positive integer greater than 2 and n is any positive integer such that 8n(m – 2) + (m – 4)2 is a square, say c2, such that the positive root c is congruent to m modulo 2(m – 2), then n is an m-gonal number. For those unfamiliar with congruences, the last condition says that c and m must differ by an integer multiple of 2(m – 2). The converse is also true.
Much to their surprise, producing this proof did not require any knowledge beyond what Krause and his classmates had learned from Skogman throughout the course.
Ozanam's mathematical publication, Récréations, published in French in 1694, was revised three times in a 150-year timespan. A second French version was revised by Jean-Étienne Montucla and published in 1778, an 1803 edition was reedited and translated into English, and a fourth edition was revised a final time and published in 1844.
"It is still a mystery to me as to why Ozanam and Montucla claimed this rule was true," said Skogman. "It was surprising [our proof] worked out so cleanly and elegantly."
To Skogman, the importance of the discovery does not lie in the result itself, but in the curiosity and investigative pursuit the project inspired in Krause. This sort of enthusiasm is often the catalyst for student research projects.
"We have a lot of curious students at Brockport, and a certain problem might pique a student's interest," said Skogman.
Krause, who will graduate with a Master of Arts in Mathematics in May, developed a strong interest in mathematics when he started taking math courses as an undergraduate physics major at Brockport. He eventually switched his major and enrolled in the Mathematics Combined Degree program.
"I began to appreciate the rigor and creativity of mathematics and eventually switched my major to math during my junior year," said Krause. "Taking proof-based courses and talking to faculty about mathematics has been the highlight of my academic career."
Krause feels Brockport has provided him "an excellent base for a PhD," and he looks forward to pursuing one of the number of programs to which he has been accepted in the fall.
Skogman and Krause submitted a paper including details about their findings to a mathematics research journal, which is awaiting word on possible publication.