An interior number in Pascal's triangle is surrounded by 6 integers.
Prove that the product of these numbers is always a square of an integer.
Each entry in a Paschal triangle can be expressed as the combination C(n,m), being n the row and m the position from the left in the row.
So we have:
C(n1,m1) C(n1,m)
C(n,m1) C(n,m) C(n,m+1)
C(n+1,m) C(n+1,m+1)
Their product ahead of C(n,m) is:
[(n1)!*n!*(n+1)!]^2

[(m1)!*m!*(m+1)!]^2*[(mn)!*(mn1)!*(mn+1)!]^2
which is always a square integer (as is the product of six integers).

Posted by armando
on 20160720 09:58:47 