Given an n &gt; 0 is it possible to partition the set <mi class="MJX-tex-caligraphic

Armeninilu

Armeninilu

Answered question

2022-06-16

Given an n > 0 is it possible to partition the set P = { 1 , 2 , , 2 n } into n pairs ( a i , b i ) such that a i + b i is a prime?

Answer & Explanation

Tianna Deleon

Tianna Deleon

Beginner2022-06-17Added 29 answers

Yes. The proof is by strong induction. The base case is obvious. By Bertrand's postulate there exists a prime p between 2 n + 1 and 4 n, so pick the pairs { p 2 n , 2 n } , { p 2 n + 1 , 2 n 1 } , . . . and so forth. Now it remains to pair up the numbers { 1 , 2 , . . . p 2 n 1 }, which is possible by the inductive hypothesis.
Brunton39

Brunton39

Beginner2022-06-18Added 8 answers

I think the way of expert partitioned the set is a bit wrong. You can create the pairs ranging from < p 2 n , 2 n > . . . < p n , n >. The remaining subset yet to be partitioned is { p ( n 1 ) , . . , 1 } which by induction you have the necessary partition.

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