Integer partitions are an active and exciting area of mathematical research producing many profound results explaining properties of the number of sums of whole numbers. The goal of this project is to develop solutions and/or record observations related to these problems.
In particular, the integer partitions of a non-negative integer n, p(n), is the number of unordered ways to sum to n using only whole numbers. A restricted partition is a rule imposed on the whole numbers used. The first conjecture is on particular sums of restricted partitions, when divided by a prime, their remainders are equivalent to the remainders p(n) when divided by the same prime.
An overpartition of n is an integer partition in which the first occurrence of a whole number in a partition can be over-lined or not. There are statistics on overpartitions which highlight their divisibility properties. An open problem is if one particular statistic, the second rank moment, is larger than another, the second crank moment, for any overpartition of n.
The project’s aim is to prove both of these conjectures and submit the proofs to publication in the appropriate academic journals.
