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 Fifth power factor (Posted on 2012-08-22)
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.

 No Solution Yet Submitted by K Sengupta No Rating

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 computer exploration | Comment 2 of 3 |

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 19687512502111    98069251575860378820108173491556    98069447549040555681007462776837 19687514470862  1786307923207969213702200270276837  1786309920930916485357408703430548 19687516439613 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

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