The Power of 2

• k has to be greater than m (otherwise $2^{k}-2^{m}$ would not be positive). This also implies that k greater than 0 (otherwise m would be negative).
• k cannot be greater than 10, because then $2^{k}-2^{m}$ would be greater than 1024. Thus, k is between 1 and 10.
• Notice that for each (k, m) pair we get a different result for $2^k - 2^m$ . Explanation: Since m is smaller than k, $2^m <= 2^{k - 1} = \frac{1}{2} 2^k$ . i.e., $2^k - 2^m$ is at least half of $2^k$ . Also, increasing k by 1 doubles $2^k$ ...
• For each value i of k, m can be between 0 and i - 1 => i options.
• Therefore, the number of possible pairs is $1 + 2 + ... + 10 = 55$ .
• Out of these, $2^{10} - 2^{0/1/2/3/4}$ would give a result that is greater than 1000, therefore we exclude these 5 pairs, leaving $\boxed{50}$ valid pairs.

Note that if $k=10$ , then $2^m>24$ so that $2^k-2^m<1000$ . Therefore $m\ge 5$

If $k\le 9$ , then $0\le m< k$ .

Since there are $k$ different possibilities for $m$ if $0\le m< k$ , then the total number of possible ways for $k\le 9$ is $1+2+\cdots +8+9=45$ .

We add the possibilities where $k=10$ . Since $5\le m< 10$ , there are $5$ ways.

Therefore, the total number of ways is $45+5=\boxed{50}$ .

m and k are non negative. 2^k - 2^m will be positive integers if k>m. m is non negative so if: m=0, k={1,2,...,9}. there are 9 solutions. m=1, k={2,3,...,9}. there are 8 solutions. m=2, k={3,4,...,9}. there are 7 solutions. m=3, k={4,5,...,9}. there are 6 solutions. m=4, k={5,6,...,9}. there are 5 solutions. m=5, k={6,7,...,10}. there are 5 solutions. m=6, k={7,8,...,10}. there are 4 solutions. m=7, k={8,9,10}. there are 3 solutions. m=8, k={9,10}. there are 2 solutions. m=9, k={10}. there is 1 solution. if m>9, then k>10, 2^k - 2^m is larger than 1000. so there is no solutions for m>9. the answer is 9+8+7+6+5+5+4+3+2+1=50.

since the positive integer required is less than 1000,

maximum value of k = 10 and corresponding values of m range from 5 to 9 value of k = 9 and corresponding values of m range from 0 to 8 value of k = 8 and corresponding values of m range from 0 to 7 value of k = 7 and corresponding values of m range from 0 to 6 value of k =6 and corresponding values of m range from 0 to 5 value of k = 5 and corresponding values of m range from 0 to 4 value of k = 4 and corresponding values of m range from 0 to 3 value of k = 3 and corresponding values of m range from 0 to 2 value of k = 2 and corresponding values of m range from 0 to 1 value of k = 1 and corresponding values of m =0

totally we get 50 numbers :)

As 2^k - 2^m should be positive, then we can make cases and calculate the total number of integers.
For m=1, k can be {2,3,4,5,6,7,8,9} m=2, k can be {3,4,5,6,7,8,9} m=3, k be {4,5,6,7,8,9} m=4, k be {5,6,7,8,9} m=5, k be {6,7,8,9,10} m=6, k be {7,8,9,10} m=7, k be {8,9,10} m=8, k be {9,10} m=9, k be {10} Adding all these cases Therefore Total number of cases = 50

The largest power of 2 below 1000 is 9 as 2^9 = 512 and 2^{10} = 1024 Fixing k = 9, m = {8,7,6 ... 2,1,0} i.e. 9 solutions as 0 < 2^9 -2^m < 1000

similarly fixing k = {8,7,6,5,4,3,2,1} we get 8,7,6,5,4,3,2,1 solutions respectively adding no. of solutions given above = 45 Now let k =10 Then 2^10 - 2^m <1000 or 1024 > 2^m > 24 It follows 2^m =32, 64, 128, 256, 512 or m = {5,6,7,8,9} 5 solutions Total solutions = 45+5 = 50

In order for our number N $2^{k}-2^{m}=N$ to be positive then $k>m$ .

So if k=1 then m has to be 0 $\longrightarrow$ 1 solution (notice the difference in the wording between non-negative and positive) if k=2 then m can either be 1 or 2.... $\longrightarrow$ 2 solutions Therefore we have k solutions for every k value. if k=10 then $2^{10}=1024$ so m can't be $2^{0}, 2^{1}, 2^{2}, 2^{3}, 2^{4}$ or otherwise N>1000

Since we have k solutions for every k value, and there are 9 values of k (plus the solutions for k=10 which are 5) ; then the number of solutions are \frac{9+1}{2}\dot 9}\=45} (sum of algebraic sequence) numer of solution from k=1 to k=9 \(\longrightarrow 45 + numer of solution from k=10 $\longrightarrow$ 5 = 50 (45+5=50)