Fractran champions

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

Champions with 0 fractions

Primes used	Max runtime	Example champion(s)	First champion's vector representation	Champion found	Notes
0	0	[]	$\left[\begin{array}{c}\end{array}\right]$

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]	$\left[\begin{array}{ccccc}0& -2& -1& 1& 1\\ 2& -1& 0& -1& 0\\ -1& 1& 2& 0& 0\\ 0& 4& -1& 2& 0\end{array}\right]$		Also see 4 fractions 19 primes.

Champions with 5 fractions

Primes used	Max runtime	Example champion(s)	First champion's vector representation	Champion found	Notes
5	1	[1/2, 1/2, 1/2, 1/2, 1/2]			See 4 fractions 4 primes.
6	2	[3/2, 1/2, 1/2, 1/2, 1/2]			See 4 fractions 5 primes.
7	3	[3/2, 5/3, 1/5, 1/2, 1/2]			See 4 fractions 6 primes.
8	5	[9/2, 5/3, 1/5, 1/2, 1/2]			See 4 fractions 7 primes.
9	7	[9/2, 25/3, 1/5, 1/2, 1/2]			See 4 fractions 8 primes.
10	11	[9/2, 25/3, 7/5, 1/7, 1/2]			See 4 fractions 9 primes.
11	16	[27/2, 25/3, 7/5, 1/7, 1/2]			See 4 fractions 10 primes.
12	28	[1/45, 4/5, 3/2, 25/3, 1/2]			See 4 fractions 11 primes.
13	58	[5/6, 7/2, 3/5, 40/7, 1/3]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 1\\ 0& 1& -1& 0\\ 3& 0& 1& -1\\ 0& -1& 0& 0\end{array}\right]$
14	137	[9/2, 7/15, 11/3, 5/7, 28/11] [7/15, 9/2, 11/3, 5/7, 28/11] [7/15, 11/3, 9/2, 5/7, 28/11]	$\left[\begin{array}{ccccc}-1& 2& 0& 0& 0\\ 0& -1& -1& 1& 0\\ 0& -1& 0& 0& 1\\ 0& 0& 1& -1& 0\\ 2& 0& 0& 1& -1\end{array}\right]$
15	746	[7/15, 22/3, 6/77, 5/2, 9/5] [7/15, 6/77, 22/3, 5/2, 9/5] [14/15, 3/77, 10/7, 11/2, 49/11]	$\left[\begin{array}{ccccc}0& -1& -1& 1& 0\\ 1& -1& 0& 0& 1\\ 1& 1& 0& -1& -1\\ -1& 0& 1& 0& 0\\ 0& 2& -1& 0& 0\end{array}\right]$		BBf(20)
16	31957632	[27/10, 5/21, 4/3, 7/2, 9/7] [7/15, 27/14, 4/3, 5/2, 9/5] [7/15, 4/3, 27/14, 5/2, 9/5]	$\left[\begin{array}{cccc}-1& 3& -1& 0\\ 0& -1& 1& -1\\ 2& -1& 0& 0\\ -1& 0& 0& 1\\ 0& 2& 0& -1\end{array}\right]$		BBf(21)
17	≥7548863488598188537	[7/30, 27/2, 8/35, 5/3, 7/5]	$\left[\begin{array}{cccc}-1& -1& -1& 1\\ -1& 3& 0& 0\\ 3& 0& -1& -1\\ 0& -1& 1& 0\\ 0& 0& -1& 1\end{array}\right]$		Probvious champion. 1 Cryptid remains: Fenrir.
18	≥113501...261856 (41 digits)	[8/15, 9/14, 25/2, 7/25, 10/7] [9/10, 49/2, 8/21, 5/49, 14/5] [9/10, 8/21, 49/2, 5/49, 14/5]	$\left[\begin{array}{cccc}3& -1& -1& 0\\ -1& 2& 0& -1\\ -1& 0& 2& 0\\ 0& 0& -2& 1\\ 1& 0& 1& -1\end{array}\right]$		Probvious champion. 2 Cryptids remain: Fenrir and 1 other Hydra-like Cryptid.
19	≥170659...455694 (69 digits)	[7/30, 27/2, 32/35, 5/3, 7/5]	$\left[\begin{array}{cccc}-1& -1& -1& 1\\ -1& 3& 0& 0\\ 5& 0& -1& -1\\ 0& -1& 1& 0\\ 0& 0& -1& 1\end{array}\right]$

Champions with 6 fractions

Primes used	Max runtime	Example champion(s)	First champion's vector representation	Champion found	Notes
15	149	[9/2, 7/15, 11/3, 5/7, 28/11, 1/5] [7/15, 9/2, 11/3, 5/7, 28/11, 1/5] [7/15, 11/3, 9/2, 5/7, 28/11, 1/5]	$\left[\begin{array}{ccccc}-1& 2& 0& 0& 0\\ 0& -1& -1& 1& 0\\ 0& -1& 0& 0& 1\\ 0& 0& 1& -1& 0\\ 2& 0& 0& 1& -1\\ 0& 0& -1& 0& 0\end{array}\right]$
16	114613...702295 (63 digits)	[1/12, 9/10, 14/3, 11/2, 5/7, 3/11] [9/10, 1/12, 14/3, 11/2, 5/7, 3/11]	$\left[\begin{array}{ccccc}-2& -1& 0& 0& 0\\ -1& 2& -1& 0& 0\\ 1& -1& 0& 1& 0\\ -1& 0& 0& 0& 1\\ 0& 0& 1& -1& 0\\ 0& 1& 0& 0& -1\end{array}\right]$		Probvious BBf(22) champion.
17	≥439325...849183 (125 digits)	[10/3, 9/14, 5/4, 121/2, 7/5, 3/11] [3/4, 25/14, 6/5, 121/2, 7/3, 5/11] [9/10, 14/3, 7/4, 121/2, 5/7, 3/11] [9/10, 7/4, 14/3, 121/2, 5/7, 3/11]	$\left[\begin{array}{ccccc}1& -1& 1& 0& 0\\ -1& 2& 0& -1& 0\\ -2& 0& 1& 0& 0\\ -1& 0& 0& 0& 2\\ 0& 0& -1& 1& 0\\ 0& 1& 0& 0& -1\end{array}\right]$		Assuming AI is correct, probvious BBf(23) champion.
18	≥900073...596451 (14449 digits)	[14/15, 1/6, 11/3, 441/2, 5/7, 15/11] [14/15, 1/6, 441/2, 11/3, 5/7, 15/11]	$\left[\begin{array}{ccccc}1& -1& -1& 1& 0\\ -1& -1& 0& 0& 0\\ 0& -1& 0& 0& 1\\ -1& 2& 0& 2& 0\\ 0& 0& 1& -1& 0\\ 0& 1& 1& 0& -1\end{array}\right]$		Best candidate for BBf(24).

Champions with 7 fractions

Primes used	Max runtime	Example champion(s)	First champion's vector representation	Champion found	Notes
16	? (known to be less than 10000)	?	?
17	≥130987...374169 (63 digits)	[1/12, 9/10, 14/3, 11/2, 5/7, 3/11, 1/5] [9/10, 1/12, 14/3, 11/2, 5/7, 3/11, 1/5]	$\left[\begin{array}{ccccc}-2& -1& 0& 0& 0\\ -1& 2& -1& 0& 0\\ 1& -1& 0& 1& 0\\ -1& 0& 0& 0& 1\\ 0& 0& 1& -1& 0\\ 0& 1& 0& 0& -1\\ 0& 0& -1& 0& 0\end{array}\right]$