# [seqfan] Re: A conjecture of Saha and Karthik.

Charles Greathouse charles.greathouse at case.edu
Thu Feb 13 15:11:02 CET 2014

```Perhaps the sequence could be

%N Pisano period of n^2 divided by Pisano period of n.
%S 1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 1, 13, 7, 15, 16, 17, 9, 19, 10, 21,
11, 23, 4, 25, 13, 27, 7, 29, 5, 31, 32, 33, 17, 35, 9, 37, 19, 39, 40, 41,
7, 43, 44, 45, 23, 47, 16, 49, 25, 17, 26, 53, 27, 55, 14, 19, 29, 59, 5,
61, 31, 63, 64, 65, 11, 67, 34, 23, 35, 71, 36, 73, 37, 75, 76, 77, 13, 79,
80, 81, 41, 83, 7, 85, 43, 87, 88, 89, 45, 13, 23, 31, 47, 95, 32, 97, 49,
99, 50, ...

with the associated conjecture
%C Conjecture (Saha & Karthik): a(n) = 1 only for n = 1, 6, and 12.

Note also that a(n) | n.

Should I submit this?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Wed, Feb 12, 2014 at 8:18 PM, L. Edson Jeffery <lejeffery2 at gmail.com>wrote:

> While reading about the Wall-Sun-Sun prime conjecture, I ran across the
> following conjecture by Saha and Karthik on the last page of their preprint
> http://arxiv.org/abs/1102.1636 :
>
>
> For n a positive integer, let pi(n) denote the least positive integer k
> such that n | F(k) and F(k+1) == 1 (mod n), where F(m) is the m-th
> Fibonacci number (pi(n) is sometimes called the 'Pisano period'). Let S
> denote the set of all solutions of pi(n^2) = pi(n) over the positive
> integers. Then S = {6,12}.
>
>
> S is reminiscent of the recent A235383 = {8,144}. Is there a deeper
> relation between the two sequences, and should the conjectured S be
> included in the database?
>
> Ed Jeffery
>
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>
```