Consider this set of numbers:
$$1, 5, 8, 11, 13, 31, 37, 53, 61, 73, 79, 97, 122, 127$$
This is the set of numbers $n$ such that $nm \pm 1$ is not a twin prime pair for all $m \leq n$.
For instance, $8$ is in this list because none of these are twin prime pairs:
$8*1 \pm 1 = (7,9)$
$8*2 \pm 1 = (15,17)$
$8*3 \pm 1 = (23,25)$
$8*4 \pm 1 = (31,33)$
$8*5 \pm 1 = (39,41)$
$8*6 \pm 1 = (47,49)$
$8*7 \pm 1 = (55,57)$
$8*8 \pm 1 = (63,65)$
This list seems to end abruptly at $127$. I've checked up to $10,000$ with this Mathematica code:
tps[a_,b_]:=ParallelTable[
Pick[n,Length@Intersection[
Select[n* Times@@@Subsets[Table[k,{k,1,n}],{1}]-1,PrimeQ]+1,
Select[n* Times@@@Subsets[Table[k,{k,1,n}],{1}]+1,PrimeQ]-1]
==0]
,{n,a,b}]
AbsoluteTiming@tps[1,10000]
(*{266.427,{1,5,8,11,13,31,37,53,61,73,79,97,122,127}}*)
Is this the end of the sequence?
How could I show this to be the end of the sequence, perhaps using some form of mathematical induction?