Powers of 2
Powers of 2
Can you select distinct numbers whose pairwise sums are powers of two? These numbers can be either positive or negative. For instance, choosing the numbers 1 and 3 results in a sum of 4, which is a power of 2.
Do you think you could do the same for four numbers? The task of finding four numbers, A, B, C, and D, such that the sums A + B, A + C, A + D, B + C, B + D, and C + D, are all powers of 2 is indeed a complex and intriguing mathematical problem. The conjecture that no such set of numbers exists remains an open question in the mathematical community, presenting a compelling challenge for mathematicians and number theorists.
We could select the numbers as -3, -1, 3 , and 5, resulting in four out of the six possible pairings summing to powers of 2:
- A + B = -3 + (-1) = -4 (not power of 2)
- A + C = -3 + 3 = 0 (not power of 2)
- A + D = -3 + 5 = 2 (which is 2^{1})
- B + C = -1 + 3 = 2 (which is 2^{1})
- B + D = -1 + 5 = 4 (which is 2^{2})
- C + D = 3 + 5 = 8 (which is 2^{1})
In this set, four pairings indeed result in powers of 2, illustrating a fascinating attempt to tackle this conjecture. However, the sums -4 and 0 are not powers of 2, leaving the conjecture unresolved. This partial solution contributes to the ongoing exploration of the problem, emphasizing the intriguing nature of the conjecture and the complexity of finding a complete solution. The journey towards potentially solving or further understanding this conjecture continues to be a captivating aspect of mathematical research and number theory.
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