Minnesota Journal of Undergraduate Mathematics

The Legendre Approximation and Arithmetic Bias

Megan Paasche, Ghaith Hiary — Mon, 19 Feb 2024 00:00:00 -0600

An interesting episode in the history of the prime number theorem concerns a formula proposed by Legendre for counting the primes below a given bound. We point out that arithmetic bias likely played an important role in arriving at that formula and in its subsequent widespread, decades-long recognition. We also show that the Legendre constant 1.08366 satisfies a certain simple and natural criterion, and conjecture that this criterion is how Legendre arrived at that erroneous constant in his formula.

Expected Value of Statistics on Type-B Permutation Tableaux

Ryan Althoff, Daniel Diethrich, Amanda Lohss, Xin-Dee Low, Emily Wichert — Mon, 19 Feb 2024 00:00:00 -0600

Type-B permutation tableaux are combinatorial objects introduced by Lam and Williams that have an interesting connection with the partially asymmetric simple exclusion process (PASEP). In this paper, we compute the expected value of several statistics on these tableaux. Some of these computations are motivated by a similar paper on permutation tableaux. Others are motivated by the PASEP. In particular, we compute the expected number of rows, unrestricted rows, diagonal ones, adjacent south steps, and adjacent west steps.

Some Thoughts on the Search for 5 × 5 and 6 × 6 Additive-Multiplicative Magic Squares

Desmond Weisenberg — Mon, 19 Feb 2024 00:00:00 -0600

An additive-multiplicative magic square is a square grid of numbers whose rows, columns, and long diagonals all have the same sum (called the magic sum) and the same product (called the magic product). There are numerous open problems about magic squares by Christian Boyer on multimagie.com. One such problem is to construct or prove the impossibility of a 5 × 5 or 6 × 6 additive-multiplicative magic square of distinct positive integers. Here, we present a possible approach to this problem and some partial results. We observe that such a square can be described by a form determined by the prime factor- izations of its entries and that identifying these forms might be helpful in finding such a square or ruling out specific magic products.