Number Theory Problem
In class March 20, Josh Zucker asked the question:
If we want to assign K colors to the initial whole numbers:
1, 2, 3, ... N
What is the largest such N where we can color the
numbers with the condition that for any 3 numbers a, b, c,
where c = a+b, at least one of these 3 number will have
a different color from the other two?
For example, if K=2, and we call the colors red and green,
we can meet this condition when N=8 with the following assignment:
Red: 1, 2, 4, 8
Green: 3, 5, 6, 7
You can check that no red number is the sum of two other red numbers,
and no green number is the sum of two other green numbers.
At the April 13 class a coloring for the numbers 1, 2, ... 22 was
presented that used only three colors.
Can you write down a K=3, N=22 coloring?
Is it possible to use only three colors for N=23 ?
If N=23 is impossible, can you prove that?
What happens with 4 colors? 5 colors?
What can you say about the largest N when K colors are available?