MathDB
ways to write 2019 as difference of two perfect squares

Source: 1st Girls in Mathematics Tournament 2019 p1 (Brazil) / Torneio Meninas na Matematica (TM^2 )

May 25, 2020
number theoryPerfect SquaresPerfect Squarecombinatorics

Problem Statement

During the factoring class, Esmeralda observed that 11, 33 and 55 can be written as the difference of two perfect squares, as can be seen: 1=12−021 = 1^2 - 0^2 3=22−123 = 2^2 - 1^2 5=32−225 = 3^2 - 2^2 a) Show that all numbers written in the form 2∗m+12 * m + 1 can be written as a difference of two perfect squares. b) Show how to calculate the value of the expression E=1+3+5+...+(2m+1)E = 1 + 3 + 5 + ... + (2m + 1). c) Esmeralda, happy with what she discovered, decided to look for other ways to write 20192019 as the difference of two perfect squares of positive integers. Determine how many ways it can do what you want.
Back to ProblemsView on AoPS