Fibonacci-Like Sequence With Mod

[Franklin T. Adams-Watters]

Neil Fernandez <primeness at borve.demon.co.uk> wrote:
>I changed the base to m^2 to get the sequence defined by
>
>a(0)= 0
>a(1)= 1
>for m>=2, a(m) = a(m-1) + a(m-2) (mod m^2)
>
>which gives
>
>1,2,3,5,8,13,21,34,55,89,0,89,89,178,11,189,200,28,228,256,0,256,...
>
>The sequence b(m), listing values of m for which a(m)=0 runs
>
>12,22
>
>Is this sequence finite? If b(3) exists, it is greater than 1 million.

It is likely to be no easier to prove that this sequence is finite than to prove that the zeros of one of the original sequences under discussion are infinite.  However, both conjectures are likely true.

Making the approximation that each value between 0 and the modulus m(n) is equally likely, the expected number of zeros in the first case (m(n) = n) is:

\Sum_{i=1}^\infinity 1/i,

which is infinite.  On the other hand, with m(n) = n^2, the expected number of zeros is:

\Sum_i=1^\infinity 1/i^2,

which is finite.

-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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