608 Infinitude of prime numbers
Project: Prime links
A prime link of lengthnis a permutation of 1, 2,.. .nbeginning with
1 and ending withnsuch that the sum of each pair of adjacent terms is
prime. This was proposed and solved by Morris Wald [157]. Forn≤ 6 ,
the link is unique. Forn=7there are two links: 1 , 4 , 3 , 2 , 5 , 6 , 7 and
and 1 , 6 , 5 , 2 , 3 , 4 , 7. Wald suggested working backwards. Start withn
and precede it with the greatest remaining member of the set whose sum
withnis a prime, and repeat in like fashion. Here are the first 10 links:
1.
1 , 2.
1 , 2 , 3.
1 , 2 , 3 , 4.
1 , 4 , 3 , 2 , 5.
1 , 4 , 3 , 2 , 5 , 6.
1 , 4 , 3 , 2 , 5 , 6 , 7.
1 , 2 , 3 , 4 , 7 , 6 , 5 , 8.
1 , 2 , 3 , 4 , 7 , 6 , 5 , 8 , 9.
1 , 2 , 3 , 4 , 7 , 6 , 5 , 8 , 9 , 10.
Continue with larger values ofn.