# סמינר תלמידי מחקר - מיכאל צ'פמן

Speaker: Michael Chapman
Title: Pell's Equation
Find all natural numbers that are both squares and triangular. Definition of Triangular Number
Find all non-negative integers a such that a+1 and 3a+1 are both perfect squares. Also, prove that if you take the ascending sequence (a_n)_{n=1}^\infty of such numbers, the number a_n \cdot a_{n+1}+1 is also a perfect square.
Prove there are infinitely many triplets of consecutive positive integers, all of them are sums of two squares. For example 8=2^2 +2^2,9=3^2 +0^2,10=3^2+1^2.
Let x be an integer. Prove that 5x^2-4 or 5x^2+4 is a perfect square if and only if x is a Fibonacci number.
In this talk we study Pell's equation, which appears naturally in all of the problems above. We first give some combinatorial and number theoretical motivation and then dive into solving the equation itself. On the path to find all solutions to Pell's equation we will encounter seeds of modern mathematical approach, connecting geometry number theory and abstract algebra.
The talk will be given in English.

## Date:

Wed, 22/11/2017 - 10:00 to 11:00

רוס 70A