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Albany, NY – In today's Academic Minute, Professor Brandt Kronholm of St. Mary's College of Maryland explains Partition Theory, and uses some very large numbers in doing so.
Brandt Kronholm is a visiting assistant professor of mathematics at St. Mary's College of Maryland where his research focuses on integer partition. He holds a Ph.D. from the University at Albany.
Prof. Brandt Kronholm - Partition Theory
Partition Theory is the branch of mathematics that studies how positive whole numbers add together, regardless of order. For example, there are only five ways that positive whole numbers can add up to 4. They are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1 and we say there are five partitions of 4.
The mathematics used to study partitions has found application to the study super cold carbon atoms, computer science, and astro-physics. The fact that there are 3 trillion 972 billion 999 million 029 thousand 388 partitions of 200 should convince us that a partition formula would be very complicated.
Though study began over 250 years ago with the blind mathematician Euler, it was not until the early 20th century that the formula was established by G. H. Hardy and the great Indian genius Ramanujan. Further, Ramanujan made the remarkable discovery that whenever a number ending in 4 or 9 is partitioned, the number of partitions is a multiple of five. There was no reason to suspect that something like this could possibly be true and has led to over 90 years of research. Only ten years ago, patterns were discovered so that the number of partitions is always a multiple of a prime, for every prime except 3.
The focus of my research is the number of parts of a partition. For instance, there are only two partitions of 4 with two parts, 3+1 and 2+2. About a year ago I discovered that given any prime, there are patterns in the integers so that the number of partitions of that integer into a prime number of parts is a multiple of that prime, including the prime 3. Euler and Ramanujan would be proud.