A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
Published December 2002,February 2002,December 2011,February 2011.
This article is about $ 2^n-n $ numbers, that is, numbers that are produced by replacing $ 'n' $ in $ 2^n-n $ with a positive integer $ (1,2,3...) $. I came across these numbers while studying Mersenne numbers $ (2^n-1) $. It got me thinking about $ 2^n-n $ numbers, if there are any interesting properties to them, and what are the properties of their primes. In the rest of the article $A_n$
will mean $ 2^n-n $. The first few numbers are: