Undergraduate Colloquium: Arian Eftekhari & Alessandro Adami

Abstract: A positive integer is called perfect if the sum of its divisors equals twice the number itself. Although the first examples, such as 6 and 28, have been known for centuries, the full structure of perfect numbers remains an open question in number theory.

In this talk, we examine the structural constraints that perfect numbers must satisfy. We show why every even perfect number must arise from a special type of prime called a Mersenne prime, and why many other numbers cannot produce a perfect number. We then turn to the case of odd perfect numbers, whose existence is still unknown, and derive necessary arithmetic conditions they must satisfy, including the requirement that any such number must be congruent to 1(mod4).

Abstract: K-nearest neighbours (KNN) is a simple yet effective supervised learning algorithm based on distance computations. However, standard KNN requires storing and repeatedly accessing the entire labeled dataset at prediction time, which can become computationally and memory intensive for large datasets. This raises a natural question: can KNN maintain strong predictive performance when restricted to only a carefully selected subset of the training data?

In this talk, we investigate this question using a large student depression dataset. We evaluate whether KNN can achieve comparable performance when trained on a well-chosen fraction of the available data, and discuss the role of multivariate imputation by chained equations in handling missing values within the dataset.