Before trying the problems "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Is 7013*2ⁿ+1 always composite?

Is 78557*2ⁿ+1 always composite?

78557*2ⁿ+1 is always composite. Checking mod (3,5,7) in Excel shows that to be prime, either 78557*2^(12n-1)+1 or 78557*2^(12n+3)+1 must be prime.

78557*2^(12n-1)+1 is always divisible by 13. The 9 smallest values of 78557*2^(12n+3)+1 have the smallest factor(s)

(37,167)

(71,73,211) 

(19) 

(37) 

(73) 

(19) 

(37) 

(73) which is enough to show that the expression is composite for all n with some factor from {3,5,7,13,19, 37,73}

7013*2ⁿ+1 is indeterminate. The same procedure gives the holdouts 7013*2^(12n+1)+1, and 7013*2^(12n+5)+1. Again, the first of these is always divisible by 13.

But the smallest factors of 7013*2^(12n+5)+1 are:

(191)
(17,37,41)
(7488079)
(17,379)
(23,37)
(17)
(41,43,193,197)
(17,37,71)
(59)
(17)

eliminating the periodic repeats gives expressions such as 7013*2^(144n-127)+1, which cannot obviously be factorised for all n.

   

Edited on September 17, 2017, 1:05 am

Posted by broll on 2017-09-17 00:53:50