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Differences 2 (Posted on 2024-03-11) Difficulty: 3 of 5
Determine the largest number of distinct positive integers you can have such that the difference between any two of them is prime.

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Some Thoughts Possible Solution | Comment 1 of 4
Assume there are 3 numbers, all even. They are distinct, so at least one pair has an even difference greater than 2, which therefore cannot be prime. (I)

Assume there are 3 numbers, all odd. They are distinct, so at least one pair has an even difference greater than 2, which therefore cannot be prime. (II)

Assume the numbers are a combination of odd and even. Then two of the numbers that are even must differ by 2, and two of the numbers that are odd must also differ by 2. By (I) above, there cannot be more than 2 such even numbers, and by (II) above, there cannnot be more than 2 such odd numbers, so the maximum possible sequence  is 4. 

4,6,9,11 is an example of such a sequence, since 11-4=7, 9-4=5, 6-4=2, 11-6=5, 9-6=3, and 11-9=2.



  Posted by broll on 2024-03-11 08:22:40
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