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Multiples of Composite Numbers Do Not Create Fractals on Pascal’s Triangle

Last week, the Mathletes proved that multiples of prime numbers create fractals (infinite complex self-similar patterns). In the process the children were exposed to the divisibility rules for prime numbers from 2, 3, 5, 7, 11, 13, and 17. The younger Mathletes focused on 2, 3, 5 and 7.

Pascal’s Triangle Prime Rows, Hexagon Sums, Fractal of Prime Multiples

One of the amazing properties of Pascal’s Triangle is that the prime rows (2,3,5,7,11,13,17,19,23,29…) are the ONLY rows of Pascal’s in which all numbers (except for the “1s”) are multiples of that prime number. For example, row 7 has 21, 35, and 42; row 11 has 55,165,330,and 462.

Pascal’s Triangle Patterns and Combinations

Now that the children have discovered the process of building the most famous array of numbers in Pascal’s Triangle, we began to discover its many patterns. 

Pascals Triangle: Diagonals of "1s", "2s", "counting", and "prime" numbers; 12, 20, and 29 Rows

Pascal’s Triangle is named after the French mathematician and philosopher Blaise Pascal (1623-1662). It is a triangular array of counting numbers. “1s” are placed along the diagonals and each other cell is the sum of the two cells above it. The largest number on the 12th row of Pascal’s Triangle is 924. The largest number on the 20th row of Pascal’s Triangle is 184,756.

Prime Pyramid

A prime number is a whole number that has exactly two factors--itself and 1. Last week, we introduced a new discovery regarding the lack of randomness of prime numbers. I have developed a challenge called a prime pyramid where each row begins with 1 and ends with the number of that row. So, row 2 begins with a 1 and ends with a 2, row 3 begins with a 1 and ends with a 3, and so on.