Counting Arithmetic Progressions Among Perfect Squares
Primary Investigator (PI) Name
Tsz Chan
Department
CSM - Mathematics
Abstract
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.
Disciplines
Algebra | Mathematics | Number Theory
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.