# Counting Arithmetic Progressions Among Perfect Squares

## Disciplines

Algebra | Mathematics | Number Theory

## Abstract (300 words maximum)

In this project, we shall characterize all the three term arithmetic progressions, (examples would be, a, a+d, a+2d, or 2, 5, 8) inside the set of perfect squares 1, 4, 9, 16, 25, … (a quadratic object). A three term arithmetic progression of perfect squares is n-a, n, n+b (formula) or 1, 25, 49 (example). We have also devised a method to count how many three term arithmetic progressions exist between two given numbers (i.e. 1-1,000,000). This has already been calculated and proven within mathematics, but for this project, we are using elementary number theory, which is much easier to grasp for non mathematicians. Pieces of elementary number theory used to reach our conclusion includes the "Inclusion-Exclusion" principle. At one point in our research, the formula includes too many numbers that are not the answer we are looking for. So we subtract from the formula to filter out correct answers, but by subtracting, we subtracted too much (-2 and -3 excludes -6), so we must add back what was accidentally subtracted twice.

## Academic department under which the project should be listed

CSM - Mathematics

## Primary Investigator (PI) Name

Tsz Chan

Counting Arithmetic Progressions Among Perfect Squares

In this project, we shall characterize all the three term arithmetic progressions, (examples would be, a, a+d, a+2d, or 2, 5, 8) inside the set of perfect squares 1, 4, 9, 16, 25, … (a quadratic object). A three term arithmetic progression of perfect squares is n-a, n, n+b (formula) or 1, 25, 49 (example). We have also devised a method to count how many three term arithmetic progressions exist between two given numbers (i.e. 1-1,000,000). This has already been calculated and proven within mathematics, but for this project, we are using elementary number theory, which is much easier to grasp for non mathematicians. Pieces of elementary number theory used to reach our conclusion includes the "Inclusion-Exclusion" principle. At one point in our research, the formula includes too many numbers that are not the answer we are looking for. So we subtract from the formula to filter out correct answers, but by subtracting, we subtracted too much (-2 and -3 excludes -6), so we must add back what was accidentally subtracted twice.