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.

   10    for N=1 to 6600000
   20      N1=N*N*N*N*N+5
   30      N2=(N+1)*(N+1)*(N+1)*(N+1)*(N+1)+5
   35      G=fnGcd(N1,N2)
   40      if G>1 then print N,N1;N2,G
   50    next N
   60   end
   70    fnGcd(A,B)
   80      Dnd=A
   90      Dvr=B
  100      repeat
  110        R=Dnd @ Dvr
  120        Dnd=Dvr:Dvr=R
  130      until R=0
  140      return(Dnd)

finds

   n             n^5 + 5                             (n+1)^5 + 5                  GCD
 533360       43162064617930483653017600005       43162469243572337970660522806 1968751
2502111    98069251575860378820108173491556    98069447549040555681007462776837 1968751
4470862  1786307923207969213702200270276837  1786309920930916485357408703430548 1968751
6439613 11073855245289742910575769522649298 11073863843522807989513632989305829 1968751

indicating that, at least up to n = 6,600,000, the GCD's are either 1 or 1,968,751, which is prime.

Posted by Charlie on 2012-08-22 15:42:14