2021 AIME I Problems/Problem 3
Find the number of positive integers less than that can be expressed as the difference of two integral powers of
We want to find the number of positive integers which can be written in the form for some non-negative integers (note that if , then ). We first observe must be at most 10; if , then . As , we can first choose two different numbers from the set in ways. This includes , , , , which are invalid as in this case. For all other choices and , the value of is less than 1000.
We claim that for all other choices of and , the values of are pairwise distinct. More specifically, if where and , we must show that . Suppose otherwise for sake of contradiction; rearranging yields . We use the fact that every positive integer has a unique binary representation:
If then ; from here we can deduce either and (contradicting the assumption that , or and (contradicting the assumption and ).
If then , and it follows that , also contradicting the assumption . Hence we obtain contradiction.*
Then there are choices for for which is a positive integer less than 1000; by the above claim, each choice of results in a different positive integer . Then there are integers which can be expressed as a difference of two powers of 2.
- Note: The uniqueness of binary representation could be rather easily proven, but if you cannot convince yourself on the spot that this is the case, consider the following alternative proof. Let where and and , for the sake of contradiction. Therefore , or . Plugging in, we see that , or , contradiction.
Note by Ross Gao
Solution 2 (Casework)
Case 1: When our answer is in the form , where is an integer such that .
We start with the subcase where it is , for some integer where (this is because the case where yields , which doesn't work because it must be a positive integer.) Note that , and . Our answer needs to be less than , so the maximum possible result (in this case) is . Our lowest result is . All the positive powers of two less than work, so we have possibilities for this subcase. For subcases and , we have and possibilities, respectively.
Case 2: When our answer is in the form of , where is an integer such that .
We can start with the subcase where . We notice that , and which is less than , so the greatest result in this subcase is actually , and the lowest is . Thus, we have possibilities. For the other four subcases, we have and possibilities, respectively.
Answer: We note that these are our only cases, as numbers in the form of and beyond are greater than .
Thus, our result is . ~jehu26
Solution 3 (Bash)
We look for all positive integers of the form where Performing casework on we can enumerate all possibilities in the table below: As indicated by the X-marks, the ordered pairs generate which are invalid.
Note that each of the remaining ordered pairs generates one unique desired positive integer.
We prove this statement as follows:
- The positive integers generated for each value of are clearly different.
- For all integers such that the largest positive integer generated for is less than the smallest positive integer generated for
Together, we have justified our claim in bold. The answer is
Solution 4 (Faster Way)
Because the difference is less than , we can simply list out all numbers that satisfy . We get , where n is an integer. Because the sequence is geometric, the difference of any two terms will be unique. will be the number of differences for . However, we also need to consider the case in which . With simple counting, we find that numbers: could be subtracted from , which makes another 5 cases. There is no need to check for higher exponents since the lowest difference would be , which exceeds .
Thus, the final answer is
Video Solution by Punxsutawney Phil
Video Solution by Power of Logic
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