Simple unsolved math problem, 5

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:

wikipedia-pascals_triangle_5

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 (x+y)^n and group the terms into like powers x^{n-k}y^k:

wikipedia-pascals_triangle_2

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: If N(a) denotes the number of times the number a > 1 appears in Pascal’s triangle then N(a) \leq 12 for all a>1.

In fact, there are no known numbers a>1 with N(a)>8 and the only number greater than one with N(a)=8 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.

One thought on “Simple unsolved math problem, 5

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

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s