Problems & Puzzles: Conjectures

Conjecture 28. Frank Buss's Conjecture

Frank Ellerman recalled my attention to some sequences (*) produced by Frank Buss and reported in the Neil Sloane's well known site.

Taken these sequences as a set, we may interpret that Frank Buss has been studying a sort of generalization of the well known Fortunate's Conjecture.

The Frank Buss's conjecture may be expressed in the following way:

If Q(n, m) = P - n! ^m and P =nxtprm( n! ^m+1), then Q= prime for all n, for m=1 to 5.

Certainly Buss knows that Q(28, 6) is composite.  Other values that I have calculated as composites are Q(6,7), Q(5,8). Then almost for sure for all m>5 we can always find some n such that Q(n, m) is composite.

Question: Can you find a counterexample to the Buss's Conjecture?

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 * See (EIS A067362,  A067363, A067364,  A067365)

On Dec 15, 2014. Dana Jacobsen wrote:

I've calculated to:

3734 m=1 OEIS A037153
2274 m=2 OEIS A067362
1733 m=3 OEIS A067363
1439 m=4 OEIS A067364
1240 m=5 OEIS A067365
 

with all results being primes.