Benchmarks of small admissible tuples

Here is a table to fill in the best upper bounds available for small admissible tuples (1~5500) and a few of others.

Rules

Efforts to fill in the blank fields in this table are very welcome. Here are the rules for the "Lucky Winner" game:

- "Winner" for each cell: You might enter each cell with a new tuple with the best upper bound and the abbreviation of your name (Your might append others' if your solution is obtained using others' work. It is encouraged to have a long chain to show the collaborative work.)

- "Lucky": Your cell will be "bold" if it is picked by an angel, if you are sufficiently lucky. The result will be qualified for showing in the main page. Your cell might be stroked through if the luck is not sustained (i.e., if the k0 claim is retracted).

Notes:

- Thomas J Engelsma have calculated upper bounds exactly for k0 up to 342, but he did not provide his tuples. So your contribution will be shown as TJE&<Your Name>, if your provide an optimal tuple.

- Update: Thomas J Engelsma has put his data online (for k0 in [2, 4199]), so there is no chance to claim optimal results for k0 up to 342 any more.

Records

[math]\displaystyle{ k_0 }[/math]	0	1	2	3	4	5	6	7	8	9
6320	-	-	-	-	-	-	-	-	-	60,726 (XFX-AVS)
5500	-	-	-	-	-	-	-	-	-	-
5450	51,498 (XFX)	51,516 (XFX)	51,520 (AVS)	51,526 (AVS)	51,534 (XFX)	51,540 (XFX&AVS)	-	-	-	-
5440	-	-	-	-	-	-	51,456 (XFX)	51,462 (XFX&AVS)	51472 (XFX&AVS)	-
4000	36,612 (XFX)	-	-	-	-	-	-	-	-	-
3000	26,606 (XFX)	-	-	-	-	-	-	-	-	-
2000	16,978 (TJE&XFX)	-	-	-	-	-	-	-	-	-
1000	7,802 (TJE&AVS)	-	-	-	-	-	-	-	-	-
340	-	-	342 (TJE&AVS)	-	-	-	-	-	-	-

Contributors

AVS: Andrew V. Sutherland

pedant

Scott Morrison

TJE: Thomas J. Engelsma

Wouter Castryck

XFX: xfxie

ZWS: Zhi-Wei Sun

Yitang Zhang: All work is based on Zhang's breakthrough paper.