We have recurrent sequence of integer number a1,a2,... a1=1, a2=2 an=3a(n-1)+5a(n-2) for n=3,4,5,... Is integer number k>=2, that a(k+1)*a(k+2) mod ak = 0 ? Please for quick help
We will show that for all this is true for now accept induction. Say then and . Now if (by contradiction) then thus . It remains to show that none of the numbers in this sequence ever are divisible by 3. Again accept induction. Say is divisible by 3 this means by induction. Which is an impossibility.