27 Hard implications
27.1 Infinite magmas
The following forward implications can be solved by ATPs, but take unusually long:
450 implies 413,426,432.
463 implies 39.
510 implies 439, 3862.
650 implies 4.
703 implies 1426,1434,4275,4307.
837 implies 107.
848 implies 835.
1119 implies 8, 99, 151, 359, 411, 1223, 1629, 1832, 2035, 3253.
1289 implies 411, 614, 1020, 2238, 2441, 2847, 3050, 4380.
1312 implies 8, 411, 1020, 1629, 1832, 2035, 3253, 3319, 3862, 3915.
The following refutations required greedy methods:
63 does not imply 1692.
73 does not imply 99, 203, 255, 4380.
118 does not imply 222, 274, 4435.
124 does not imply 1924.
476 does not imply 359, 3862, 4065.
503 does not imply 359, 3862, 4065.
511 does not imply 614, 714, 1020, 1223, 2238, 2338, 2441, 2847, 3050, 4380, 4435.
677 does not imply 255.
707 does not imply 1223, 1316, 2238, 2847, 2940.
879 does not imply 4065.
883 does not imply 2035, 2101, 2238, 2304.
906 does not imply 3862.
1076 does not imply 47, 99, 151, 203, 255, 411, 614, 817, 1223, 1426, 1629, 1832, 2035, 2238, 2441, 2644, 2847, 3050, 3253, 3456, 3659, 3862, 4065, 4380.
1112 does not imply 8, 411, 1629, 1832, 3253, 3319, 3862, 3915.
1113 does not imply 2441, 2534.
1289 does not imply 3116, 4815.
1648 does not imply 3253.
1692 does not imply 47, 1832, 2441, 3050, 3456, 4065.
The following refutations required ad hoc methods:
Here is a list of the final implications to refute that required analysis in the project:
Three-variable equations:
1485 does not imply 151, 3456, 3862.
854 does not imply 413, 1045, 1055, 3316, 3925.
1117 does not imply 2441.
1437 does not imply 4269.
The “Hardy–Ramanujan cluster”
917 does not imply 1629, 1729, 2441, 2541.
1323 does not imply 2744.
1526 does not imply 1223, 2744.
1729 does not imply 817, 917.
2541 does not imply 1729.
2744 does not imply 1323, 1526.
Other two-variable equations:
1722 does not imply 644, 3050, 1832.
1516 does not imply 255.
1518 does not imply 47, 614, 817, 3862.
3308 does not imply 3456.
3342 does not imply 3456, 3522, 4065, 4118.
3352 does not imply 4065.
3475 does not imply 3659.
27.2 Finite magmas
In the case of finite magmas, there are some non-trivial positive implications:
467 implies 2847.
906 implies 3862.
1516 implies 255.
1441 implies 4067.
1443 implies 3055.
1681 implies 3877.
1701 implies 1035.
1113 implies 1167.
1167 implies 1096.
There are also some tricky negative implications:
476 does not imply 359, 4065.
503 does not imply 4065, 3862.
854 does not imply 107, 1248, 1251.
1076 does not imply 2234, 2294, 4435.
1518 does not imply 47, 817.
3352 does not imply 4065.
3475 does not imply 3659.
Finally, we have some open questions: