Determine the largest number of distinct positive integers you can have such that the difference between any two of them is prime.
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 114=7, 94=5, 64=2, 116=5, 96=3, and 119=2.

Posted by broll
on 20240311 08:22:40 