2. Bertrand's postulate (now known as a theorem) states that for every number N>1, there exists a prime number between N and 2N. Use this to show that every integer greater that 6 can be written as the sum of one or more distinct prime numbers.
For example: 7=7, 8=3+5, 9=2+7,
10=3+7 (or 10=2+3+5), 11=11, etc.
Sept. 15 -
1. The integers from 0 to 104 inclusive are written clockwise on a circle. (It's a 105 hours clock!). A frog starts at 0 and jumps clockwise around the circle by a fixed difference d, until he ends up back at 0. For example, if d=35, he would jump to 35, 70 and then back to 0. For how many positive integers d such that d<105 will the frog jump onto every number before returning to 0?
2. (Josephus Problem, 1st century AD) Ten thousands sailors are arranged around the edge of their ship and they are numbered 1, 2, 3, ..., 10000. Starting the count with number 1, every other sailor is pushed overboard (by the mad captain) until they are all gone. Where should you be standing to be the last survivor.
||Arithmetic & Geometric
|Oct. 13 - no meeting
for Big Circle
|Little Circle meets this day
||Telescopic Sums and Products
||The Pigeon Hole Principle