Fractran champions

In this article we focus on the longest running Fractran programs containing a specific number of fractions and primes.

Champions with 1 fraction

Primes used	Max runtime	Example champion(s)	First champion's vector representation	Champion found	Notes
1	1	[1/2]	$\left[\begin{array}{c}-1\end{array}\right]$
2	1	[3/2]	$\left[\begin{array}{cc}-1& 1\end{array}\right]$
3	1	[9/2]	$\left[\begin{array}{cc}-1& 2\end{array}\right]$
4	1	[27/2]	$\left[\begin{array}{cc}-1& 3\end{array}\right]$
n	1	$[3^{n-1}/2]$	$\left[\begin{array}{cc}-1& n-1\end{array}\right]$

Champions with 2 fractions

Primes used	Max runtime	Example champion(s)	First champion's vector representation	Champion found	Notes
2	1	[1/2, 1/2]	$\left[\begin{array}{c}-1\\ -1\end{array}\right]$
3	2	[3/2, 1/3] [1/3, 3/2]	$\left[\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right]$
4	3	[9/2, 1/3] [1/3, 9/2]	$\left[\begin{array}{cc}-1& 2\\ 0& -1\end{array}\right]$
5	4	[27/2, 1/3] [1/3, 27/2]	$\left[\begin{array}{cc}-1& 3\\ 0& -1\end{array}\right]$
6	5	[81/2, 1/3] [1/3, 81/2]	$\left[\begin{array}{cc}-1& 4\\ 0& -1\end{array}\right]$
7	6	[243/2, 1/3] [1/3, 243/2]	$\left[\begin{array}{cc}-1& 5\\ 0& -1\end{array}\right]$
8	7	[729/2, 1/3] [1/3, 729/2]	$\left[\begin{array}{cc}-1& 6\\ 0& -1\end{array}\right]$
9	8	[2187/2, 1/3] [1/3, 2187/2]	$\left[\begin{array}{cc}-1& 7\\ 0& -1\end{array}\right]$
n	n-1	$[3^{n-2}/2,1/3]$ $[1/3,3^{n-2}/2]$	$\left[\begin{array}{cc}-1& n-2\\ 0& -1\end{array}\right]$

Champions with 3 fractions

Primes used	Max runtime	Example champion(s)	First champion's vector representation	Champion found	Notes
3	1	[1/2, 1/2, 1/2]	$\left[\begin{array}{c}-1\\ -1\\ -1\end{array}\right]$
4	2	[3/2, 1/3, 1/2] [1/3, 3/2, 1/2]	$\left[\begin{array}{cc}-1& 1\\ 0& -1\\ -1& 0\end{array}\right]$
5	3	[9/2, 1/3, 1/2] [3/2, 5/3, 1/5] [15/2, 1/3, 1/5]	$\left[\begin{array}{cc}-1& 2\\ 0& -1\\ -1& 0\end{array}\right]$
6	5	[9/2, 5/3, 1/5]	$\left[\begin{array}{ccc}-1& 2& 0\\ 0& -1& 1\\ 0& 0& -1\end{array}\right]$
7	7	[9/2, 25/3, 1/5] [27/2, 5/3, 1/5]	$\left[\begin{array}{ccc}-1& 2& 0\\ 0& -1& 2\\ 0& 0& -1\end{array}\right]$
8	10	[27/2, 25/3, 1/5]	$\left[\begin{array}{ccc}-1& 3& 0\\ 0& -1& 2\\ 0& 0& -1\end{array}\right]$
9	13	[27/2, 125/3, 1/5] [81/2, 25/3, 1/5]	$\left[\begin{array}{ccc}-1& 3& 0\\ 0& -1& 3\\ 0& 0& -1\end{array}\right]$
10	17	[81/2, 125/3, 1/5]	$\left[\begin{array}{ccc}-1& 4& 0\\ 0& -1& 3\\ 0& 0& -1\end{array}\right]$
11	21	[81/2, 625/3, 1/5] [243/2, 125/3, 1/5]	$\left[\begin{array}{ccc}-1& 4& 0\\ 0& -1& 4\\ 0& 0& -1\end{array}\right]$
12	26	[243/2, 625/3, 1/5]	$\left[\begin{array}{ccc}-1& 5& 0\\ 0& -1& 4\\ 0& 0& -1\end{array}\right]$
2n	$\ge n^2-2n+2$	$[3^{n-1}/2,5^{n-2}/3,1/5]$	$\left[\begin{array}{ccc}-1& n-1& 0\\ 0& -1& n-2\\ 0& 0& -1\end{array}\right]$
2n+1	$\ge n^2-n+1$	$[3^{n-1}/2,5^{n-1}/3,1/5]$	$\left[\begin{array}{ccc}-1& n-1& 0\\ 0& -1& n-1\\ 0& 0& -1\end{array}\right]$