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2018-04-05
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Anna Loria
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
odd.
"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.
