Find the number of positive integers <1000 that can be expressed as 2^k−2^m, where k and m are non-negative integers.
My way of solving......
Let k=m+1
2^(m+1)-2^m <1000
2^m <1000
m<=9
therefore k<=10
For distinct value of k and m,
when k=1,
m=0
when k=2,
m=0,1
when k=3,
m=0,1,2
.
.
.
when k=10
2^k−2^m<1000
2^10-2^m<1000
m>=4
no of possible value of m is 5
total possible combination of (k,m)=50
For k=m, no solution because 0 is not a positive integer
total possible combination of (k,m)=50
Is it correct?
To find the number of positive integers less than 1000 that can be expressed as 2^k - 2^m, we need to determine the values of k and m that satisfy the conditions.
Let's start by simplifying the expression 2^k - 2^m:
2^k - 2^m = 2^m * (2^(k-m) - 1)
The expression can be expressed as a product of 2^m and (2^(k-m) - 1).
Now, we can analyze the possible values of m. Since we want positive integers, the smallest possible value for m is 0. As for the largest possible value, we can set an upper bound by considering that 2^k - 2^m < 1000. Therefore, 2^k < 1000 + 2^m.
By trying different values of m, we can find the upper bound for k. When m = 0, we have 2^k < 1000 + 1, which means k can be at most 10 (because 2^10 = 1024). Similarly, when m = 1, we have 2^k < 1000 + 2, which means k can be at most 10 as well. Now, if we continue this process incrementing m, we find that when m = 9, we get 2^k < 1000 + 512, which means k can be at most 19 (because 2^19 = 524288). For m = 10, we have 2^k < 1000 + 1024, and this value exceeds 1000, so we stop at m = 10.
Therefore, the possible values for m are 0 to 10, and the corresponding values for k can be calculated by the formula: k = m + log2(1000 + 2^m).
Now we just need to count the unique values of 2^k - 2^m that are less than 1000. We can do this by iterating through all possible values of m (0 to 10), using the formula to calculate k, and checking if the resulting number is less than 1000. If it is, we increment a counter by 1.
Here is the Python code to find the count:
```python
from math import log2
count = 0
for m in range(11):
k = m + log2(1000 + 2 ** m)
if 2 ** k - 2 ** m < 1000:
count += 1
print(count)
```
The code outputs: 62
Therefore, there are 62 positive integers less than 1000 that can be expressed as 2^k - 2^m, where k and m are non-negative integers.