A Number Theory Quest For Oppenheimer and Psi Numbers

This paper solves two problems from two different journals by the MAA, Oppenheimer numbers from Math Horizons and Psi numbers from The College Mathematics Journal. Both of these problems were tackled similarly. First, we would attempt to solve the problem by hand using the rules provided to us. Then if our dataset was too small or if making the numbers was too time consuming, we would optimize the process in some way to broaden our dataset. From there we were able to notice patterns in the numbers and create rules to make numbers outside our dataset. We first started out with Oppenheimer numbers, a unique class of 2n-digit numbers meeting specific greatest common divisor and divisibility criteria. Using Python, we identified all 4-digit Oppenheimer numbers and extended our search to 6-digit and 8-digit Oppenheimer numbers, confirming the existence of many such numbers. We further proved that for every n ≥ 2, there is at least one 2n-digit Oppenheimer number. As the Oppenheimer problem proved straightforward enough after we had written the code we worked on the Psi problem next. By using mathematical induction, we proved whether it is possible for all integers n to be written as n distinct powers of ψ given ψ^a + ψ^b = 2 and ψ > 0. Using various rules and patterns we were able to identify a representation of the numbers 2 to 9 in the desired form and proved that such a representation exists for all integers.