Some algebraical properties of Pascal's triangle
by
Peter Panteleev
Moscow State University, Russia

In 1653 a french mathematician named Blaise Pascal described a triangular arrangment of numbers corresponding to the number of ways to choose n objects from a group of m indistinguishable objects. At the tip of Pascal's triangle is the number 1 (the zeroth row). The first row contains two 1's, both formed by adding the two numbers above them to the left and to the right, in this case 1 and 0 (all numbers outside the triangle are 0's). Do the same to create the second row 0+1=1, 1+1=2, 1+0=1. In this way the rows of the Pascal's triangle go on to infinity.

Consider the sums of numbers taken from diagonals of Pascal's triangle: u1=1, u2=1, u3=1+1=2, u4=1+2=3, u5=1+3+1=5, u6=1+4+3=8 ... . We can prove that every number ui, except u1, u2 and u6, has a proper prime divisor (the prime number p is called a proper prime divisor of ui, if p divides ui and p does not divides uk for all k < i). We consider also some Polynomials from Pascal's triangle and some special graphs, connected with the triangle.

Date received: March 9, 2001