M is a positive integer such that 4^M + 2^M +1 is a prime number.

Can there be any instance where M is not a power of 3?

Give reasons for your answer.

(In reply to re: Solution by Jer)

I hadn't interpreted it like that, e.g.

2^6 is a 'power of 3' in the sense that it is also a cube, which is a 'power of 3'

Small (ish) solutions for n=3m include:

3  73 P 
6  4161 3x19x73 
9  262657 P 
12  16781313 3×19×37×73×109 
15  1073774593 73×631×23311 
18  68719738881 3×87211×262657 
21  4398048608257 73×92737×649657 
24  '281474993487873 3×19×37×73×109×433×38737 
27  18014398643699713 2593×71119×97685839 
30  1152921505680588801 3×19×73×631×23311×18837001 
33  73786976303428141057 73×199×153649×33057806959 
36  4722366482938364690433 3×87211×246241×262657×279073 
39  302231454904207049490433 73×937×6553×86113×7830118297 
42  19342813113838464841809921 3×19×73×92737×649657×77158673929

I'd be quite surprised if any higher power was prime however 'power of 3' is interpreted.

Edited on January 21, 2016, 9:23 am

Footnote: 4^81+2^81+1 has 487 as a factor.

Edited on January 21, 2016, 9:34 am

Posted by broll on 2016-01-21 09:21:05