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]	$\left[\begin{array}{c}-1\end{array}\right]$		BBf(2)
2	1	[3/2]	$\left[\begin{array}{cc}-1& 1\end{array}\right]$		BBf(3)
3	1	[9/2]	$\left[\begin{array}{cc}-1& 2\end{array}\right]$		BBf(4)
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]	$\left[\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right]$		BBf(5)
4	3	[9/2, 1/3] [1/3, 9/2]	$\left[\begin{array}{cc}-1& 2\\ 0& -1\end{array}\right]$		BBf(6)
5	4	[27/2, 1/3] [1/3, 27/2]	$\left[\begin{array}{cc}-1& 3\\ 0& -1\end{array}\right]$		BBf(7)
6	5	[81/2, 1/3] [1/3, 81/2]	$\left[\begin{array}{cc}-1& 4\\ 0& -1\end{array}\right]$		BBf(8)
7	6	[243/2, 1/3] [1/3, 243/2]	$\left[\begin{array}{cc}-1& 5\\ 0& -1\end{array}\right]$		BBf(9)
8	7	[729/2, 1/3] [1/3, 729/2]	$\left[\begin{array}{cc}-1& 6\\ 0& -1\end{array}\right]$		BBf(10)
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]	$\left[\begin{array}{ccc}-1& 3& 0\\ 0& -1& 2\\ 0& 0& -1\end{array}\right]$		BBf(11)
9	13	[27/2, 125/3, 1/5] [81/2, 25/3, 1/5]	$\left[\begin{array}{ccc}-1& 3& 0\\ 0& -1& 3\\ 0& 0& -1\end{array}\right]$		BBf(12)
10	17	[81/2, 125/3, 1/5]	$\left[\begin{array}{ccc}-1& 4& 0\\ 0& -1& 3\\ 0& 0& -1\end{array}\right]$		BBf(13)
11	21	[81/2, 625/3, 1/5] [243/2, 125/3, 1/5]	$\left[\begin{array}{ccc}-1& 4& 0\\ 0& -1& 4\\ 0& 0& -1\end{array}\right]$		BBf(14)
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	$\ge n^2-n+1$	$[3^{n-1}/2,5^{n-1}/3,1/5]$	$\left[\begin{array}{ccc}-1& n-1& 0\\ 0& -1& n-1\\ 0& 0& -1\end{array}\right]$

Champions with 4 fractions

Primes used	Max runtime	Example champion(s)	First champion's vector representation	Champion found	Notes
4	1	[1/2, 1/2, 1/2, 1/2]	$\left[\begin{array}{c}-1\\ -1\\ -1\\ -1\end{array}\right]$
5	2	[3/2, 1/3, 1/2, 1/2]	$\left[\begin{array}{cc}-1& 1\\ 0& -1\\ -1& 0\\ -1& 0\end{array}\right]$
6	3	[3/2, 5/3, 1/5, 1/2] [15/2, 1/3, 1/5, 1/2] [9/2, 1/3, 1/2, 1/2]	$\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 0& 0& -1\\ -1& 0& 0\end{array}\right]$
7	5	[9/2, 5/3, 1/5, 1/2]	$\left[\begin{array}{ccc}-1& 2& 0\\ 0& -1& 1\\ 0& 0& -1\\ -1& 0& 0\end{array}\right]$
8	7	[9/2, 25/3, 1/5, 1/2] [27/2, 5/3, 1/5, 1/2] [9/2, 5/3, 7/5, 1/7] [9/2, 35/3, 1/5, 1/7]	$\left[\begin{array}{ccc}-1& 2& 0\\ 0& -1& 2\\ 0& 0& -1\\ -1& 0& 0\end{array}\right]$
9	11	[9/2, 25/3, 7/5, 1/7]	$\left[\begin{array}{cccc}-1& 2& 0& 0\\ 0& -1& 2& 0\\ 0& 0& -1& 1\\ 0& 0& 0& -1\end{array}\right]$
10	16	[27/2, 25/3, 7/5, 1/7]	$\left[\begin{array}{cccc}-1& 3& 0& 0\\ 0& -1& 2& 0\\ 0& 0& -1& 1\\ 0& 0& 0& -1\end{array}\right]$
11	28	[1/45, 4/5, 3/2, 25/3]	$\left[\begin{array}{ccc}0& -2& -1\\ 2& 0& -1\\ -1& 1& 0\\ 0& -1& 2\end{array}\right]$		BBf(15)
12	53	[1/45, 4/5, 3/2, 125/3]	$\left[\begin{array}{ccc}0& -2& -1\\ 2& 0& -1\\ -1& 1& 0\\ 0& -1& 3\end{array}\right]$		BBf(16)
13	107	[5/6, 49/2, 3/5, 40/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 3& 0& 1& -1\end{array}\right]$		BBf(17)
14	211	[5/6, 49/2, 3/5, 80/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 4& 0& 1& -1\end{array}\right]$		BBf(18)
15	370	[5/6, 49/2, 3/5, 160/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 5& 0& 1& -1\end{array}\right]$		BBf(19)
16	596	[5/6, 49/2, 3/5, 320/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 6& 0& 1& -1\end{array}\right]$
17	3994	[7/45, 20/21, 375/2, 9/5]	$\left[\begin{array}{cccc}0& -2& -1& 1\\ 2& -1& 1& -1\\ -1& 1& 3& 0\\ 0& 2& -1& 0\end{array}\right]$
18	60133	[1/45, 80/21, 75/2, 63/5]	$\left[\begin{array}{cccc}0& -2& -1& 0\\ 4& -1& 1& -1\\ -1& 1& 2& 0\\ 0& 2& -1& 1\end{array}\right]$
19	270858044796	[7/45, 4/21, 75/2, 3969/5]	$\left[\begin{array}{cccc}0& -2& -1& 1\\ 2& -1& 0& -1\\ -1& 1& 2& 0\\ 0& 4& -1& 2\end{array}\right]$
20	≥270858044796	[77/45, 4/21, 75/2, 3969/5]	$\left[\begin{array}{ccccc}0& -2& -1& 1& 1\\ 2& -1& 0& -1& 0\\ -1& 1& 2& 0& 0\\ 0& 4& -1& 2& 0\end{array}\right]$		Essentially the same as 4 fractions 19 primes.