Determine the smallest positive integer X, such that X is not expressible in the form (2^{P} – 2^{Q})/(2^{R} – 2^{S}), where each of P, Q, R and S is a positive integer.
Let K = (2^P  2^Q)/(2^R  2^S)
=> K = 2^Q (2^(PQ)1)/ {2^S (2^(RS)1)}
=> K = 2^(QS) (2^(PQ)1)/(2^(RS)1)
=> K = 2^a (2^b  1)/(2^c  1)
For this to be integer, b should be multiple of c
=> K = 2^a (2^cx  1)/(2^c  1)
If K is not expressible, then K/2^a is also not expressible
The first integer not expressible is an odd number
=> K = 1 + 2^c + 2^2c + .... + 2^(x1)c
The first odd number not expressible as above is the required
3 = 1 + 2^1
5 = 1 + 2^2
7 = 1 + 2^2 + 2^3
9 = 1 + 2^4
11 = 1 + 2 + 2^3 (not in GP)
So, 11 is the smallest integer

Posted by Praneeth
on 20100108 10:12:37 