It seems everyone’s heard of Pascal’s triangle. However, if you haven’t then it is an infinite triangle of integers with *1*‘s down each side and the inside numbers determined by adding the two numbers above it:

First 6 rows of Pascal’s triangle

The first 6 rows are depicted above. It turns out, these entries are the binomial coefficients that appear when you expand and group the terms into like powers :

First 6 rows of Pascal’s triangle, as binomial coefficients.

The history of Pascal’s triangle pre-dates Pascal, a French mathematician from the 1600s, and was known to scholars in ancient Persia, China, and India.

Starting in the mid-to-late 1970s, British mathematician David Singmaster was known for his research on the mathematics of the Rubik’s cube. However, in the early 1970’s, Singmaster made the following conjecture [1].

**Conjecture**: I*f denotes the number of times the number appears in Pascal’s triangle then for all .*

In fact, there are no known numbers with and the only number greater than one with is *a=3003*.

References:

[1] Singmaster, D. “Research Problems: How often does an integer occur as a binomial coefficient?”, *American Mathematical Monthly*, **78**(1971) 385–386.

### Like this:

Like Loading...

*Related*

Pingback: Simple unsolved math problem, 5 | Guzman's Mathematics Weblog