[Guest]

Determine the minimum value of a positive integer N, such that N is not expressible in the form (2^A – 2^B)/ (2^C – 2^D), where each of A, B, C and D is a positive integer.

Edited October 21, 2009 by K Sengupta

[superprismatic]

9

Edited October 21, 2009 by superprismatic

[Guest]

9

Not quite, for:

(2⁷ - 2¹)/(2⁴ - 2¹) = 9

Edited October 22, 2009 by methinks

[Guest]

11

[Guest]

Every number that can be expressed as (2^A - 2^B)/(2^C - 2^D) can also be expressed as:

2^k.(2^m - 1)/(2ⁿ - 1) -- (1)

where m, n, k are integers with k >= 0, m > 0, n > 0 and m > n.

For every positive integer of the form 2^a.X (where X is odd) that can not be expressed in the equation of the form (1), X can not also be expressed in the same form. So we can simplify by assumming that the number 'N' that can't be expressed in the form (1) is odd.

With this, k = 0 in (1)

Let,

X = (2^m - 1)/(2ⁿ - 1)

=> 2ⁿ(2^m-n - 1) = (X - 1)(2ⁿ - 1)

Let, p = m - n and Y = X - 1,

=> 2ⁿ(2^p - 1) = Y(2ⁿ - 1) -- (2)

We now have a few cases

Case I

If Y is a power of 2, we can always get values for p and n in (2).

Case II

If Y is of the form 2.Q, where Q is odd

From (2), n = 1 and (2^p - 1) = Q.

So if 'Q' can't be expressed as (2^p - 1), then we can get a value for 'N'.

First such value for 'Q' is 5 => Y = 2.Q = 10 => X = Y + 1 = 11

Multiplying this by powers of 2 will yeild other values that can not be expressed in the given form: 22, 44, 88, ..

Case II

If Y is of the form 2^r.Q, where Q is odd and r > 1

From (2), n = r, (2^p - 1) = Q(2^r - 1)

which brings us back to the original form itself.

Or in other words, for every odd number 'Q' than can't be expressed as in (1),

2^r.Q + 1 also can't be expressed (r > 1) in the same form.

So, numbers like 4*11 + 1 = 45, 89 etc also can't be expressed in that form.

Edited October 22, 2009 by methinks