Article 6RJ1G Squares, triangles, and octal

Squares, triangles, and octal

by
John
from John D. Cook on (#6RJ1G)

I ran across the following theorem in Ross Honsberger's book Mathematical Morsels:

Every odd square ends in 1 in base 8, and if you cut off the 1 you have a triangular number.

A number is an odd square if and only if it is the square of an odd number, so odd squares have the form (2n + 1)^2.

Both parts of the theorem above follow from the calculation

( (2n + 1)^2 - 1 ) / 8 = n(n + 1) / 2.

In fact, we can strengthen the theorem. Not only does writing the nth odd square in base 8 and chopping off the final digit give some triangular number, it gives the nth triangular number.

The post Squares, triangles, and octal first appeared on John D. Cook.
External Content
Source RSS or Atom Feed
Feed Location http://feeds.feedburner.com/TheEndeavour?format=xml
Feed Title John D. Cook
Feed Link https://www.johndcook.com/blog
Reply 0 comments