Page started - 12/2/99
Last updated - 20/6/99
Answer 11: Amnesty (part 2)
Cell numbers must be of the form 2n i.e. 1, 2, 4, 8, 16, 32, ...Reasoning
We require all numbers that can not be expressed as the sum of 2 or more consecutive integers i.e.N = n + (n+1) + (n+2) + ... + m = (m+n)(m-n+1)/2
For all odd numbers we can choose m=(N+1)/2 and n=(N-1)/2.
Since when m+n is even, m-n+1 is odd and vice versa, N cannot be of the form 2a.
All other even numbers can be expressed as a product of prime factors,
N=2apbqc...
We can thus choose m=N/p +(p-1)/2 and n=|(N/p) - (p-1)/2|