Let n be a positive integer > 1, such that n⁵+5 and (n+1) ⁵+5 have a positive common factor, m. Find the possible values of m.

(In reply to computer exploration by Charlie)

Yes, because the periods are:

n = 1968751k+96502 
n = 1968751k+533360  
n = 1968751k+533361 
n = 1968751k+1066696 
n = 1968751k+1707583  

and 533361= 533360+1. So my 'Tricks alert!' antenna were working after all.

Incidentally, 61031281=1968751*31 has no less than 25 such periods, although none are consecutive.

Posted by broll on 2012-08-23 01:39:47