In his 1798 number theory textbook titled Disquisitiones Arithmeticae, Carl Friedrich Gauss stated a formula for the average class number for negative discriminants. Our goal was to find a formula for the average class number of Almost Pythagorean Triples (APT’s). A Pythagorean triple is a set of 3 integers, a, b, c, that satisfies a2+b2=c2. An almost Pythagorean triple is a set of 4 integers, a, b, c, D, that satisfies a2+b2=c2+D. In this project, we strictly studied the case where D<0. APT’s are separated into classes based on their D value. For example, the triples (3, 0, 4) and (1, 1, 3) both have a D value of -7, so they are in the same class. As they are the only two triples with this D value, the class number of -7, h(-7) = 2. We wrote a code that, for every D value within the inputted range, found every APT and the average class number. This would serve as a way to guide and test our future formula. We attempted several methods for constructing an APT average class number formula. Firstly, we looked at this problem geometrically, using two paraboloids, one using an arbitrarily large N value (where N=|D|), x2+y2=z2-N and one being the smallest possible N value, x2+y2=z2-1. We set up a double integral to find the volume between the two surfaces. We tried many methods to solve this integral unsuccessfully, which ultimately led us to switch course and attempt a double summation that uses lattice counting to sum up all APT’s for an arbitrary N value: ∑∑sqrt(x2+y2+N) – ∑∑x+y. We used Euler’s summation formula to simplify this double summation into a double integral problem. Using Big O notation, we were able to ignore terms that would become insignificant as N approaches infinity. Ultimately, we found that the average class number of APT’s can be approximated using 𝝅/6 sqrt(N) where N=|D|. This finding is significant because it proves a previously unknown fact about almost Pythagorean triples and it is a viable extension of Gauss’ work with average class numbers. This project could be extended by doing similar work with almost Pythagorean Quadruples or Eisenstein Triples.
In Spring 2022, we embarked on a mission to mirror Gauss’s work with binary quadratic forms. Gauss created a way to classify binary quadratic forms based on their (negative) discriminant values, then proceeded to find a formula to calculate the average class number of binary quadratic forms with negative discriminant values. We wanted to take this concept and apply it to Almost Pythagorean Triples (APTs).
An APT is a set of 4 integers, a, b, c, D, that satisfies a2+b2=c2+D. In this project, we strictly studied the case where D<0, similar to how Gauss studied binary quadratic forms with negative discriminants. We separated APTs into classes based on their (negative) D value. For example, the triples (3, 0, 4) and (1, 1, 3) both have a D value of -7, so they are in the same class. As they are the only two reduced triples with this D value, the class number of -7, h(-7) = 2.
