Difference between revisions of "Williams prime MP least"

Latest revision as of 00:47, 8 September 2024

Description

Here are shown the least n-value ≥ 1 for any base b with 2 ≤ b ≤ 1024 which generates a Williams prime of the form (b-1)•bⁿ+1.

History

2024-08-13: Found Williams 437•438¹²⁶⁸⁰⁴+1, Predrag Kurtovic
2024-06-19: Found Williams 541•542¹⁵⁶⁶⁸⁰+1, Lleyton Joseph
2024-05-11: Found Williams 122•123⁸⁶⁵⁸⁹⁰+1, Predrag Kurtovic
2019-08-06: Found Williams 633•634⁸⁴⁸²²+1, Karsten Bonath
2019-06-20: Found Williams 952•953⁶⁰⁹⁹⁴+1, Karsten Bonath
2019-06-15: Found Williams 1003•1004²⁹⁶⁸⁴+1, Karsten Bonath
2019-06-14: Found Williams 973•974¹⁶²²+1, Karsten Bonath
2019-06-13: Found Williams 922•923⁴⁴⁷⁰+1, Karsten Bonath
2019-06-11: Found Williams 949•950⁵⁹⁵⁶+1, Karsten Bonath
2019-06-09: Found Williams 889•890³⁷³⁷⁶+1, Karsten Bonath
2019-06-06: Found Williams 868•869¹²²⁸⁸+1, Karsten Bonath
2019-06-06: Found Williams 838•839²¹⁷²+1, Karsten Bonath
2019-06-06: Found Williams 835•836¹⁰⁴⁸+1, Karsten Bonath
2019-06-03: Found Williams 826•827⁸⁴⁵²+1, Karsten Bonath
2019-06-02: Found Williams 796•797⁵¹⁸⁸+1, Karsten Bonath
2019-06-02: Found Williams 787•788¹³⁵⁴⁰+1, Karsten Bonath
2019-05-31: Found Williams 781•782¹²⁰⁰+1, Karsten Bonath
2019-05-30: Found Williams 592•593¹⁶⁶⁰+1, Karsten Bonath
2019-05-30: Found Williams 577•578²⁷⁶⁴+1, Karsten Bonath
2019-05-30: Found Williams 571•572¹⁸³²+1, Karsten Bonath
2019-05-29: Found Williams 757•758³⁰¹⁴+1, Karsten Bonath
2019-05-28: Found Williams 563•564³⁸⁰⁶⁴+1, Karsten Bonath
2019-05-25: Found Williams 673•674⁶³⁴²+1, Karsten Bonath
2019-05-25: Found Williams 700•701⁸⁶⁶²+1, Karsten Bonath
2019-05-23: Found Williams 544•545¹⁸²²+1, Karsten Bonath
2019-05-22: Found Williams 535•536⁹⁷⁶²+1, Karsten Bonath
2019-05-22: Found Williams 496•497⁴⁰¹²+1, Karsten Bonath
2019-05-10: Found Williams 362•363¹⁴²⁸⁷⁶+1, Karsten Bonath
2018-06-28: Found Williams 451•452⁷¹⁹⁴⁰+1, Karsten Bonath
2018-06-13: Found Williams 250•251¹⁰²⁹⁷⁸+1, Karsten Bonath
2018-05-15: Found Williams 325•326⁶⁴⁷⁵⁶+1, Karsten Bonath
2018-05-14: Found Williams 346•347⁶⁹⁶⁶⁰+1, Karsten Bonath
2018-04-27: Found Williams 201•202⁴⁶⁷⁷³+1, Karsten Bonath
2018-04-23: Found Williams 297•298⁶⁰⁶⁷⁰+1, Karsten Bonath

Wanted values [searched range]

There are 18 sequences:

342 [300000]
362 [304000]
422 [300000]
479 [539000]
487 [600000]
512 [500000]
602 [200000]
757 [100000]
767 [100000]
817 [100000]
830 [100000]
872 [100000]
893 [100000]
932 [200000]
992 [200000]
997 [100000]
1005 [100000]
1007 [100000]

Notes:

Base 512: Proth 511•2ⁿ+1 (first n-value divisible by 9).
Bases 342, 487, 757, 997, 1005 are sometimes reserved and worked on by Conjectures 'R Us.

Data

The data file can be found here.

Base	`n`-value
2	1
3	1
4	1
5	2
6	1
7	1
8	2
9	1
10	3
11	10
12	3
13	1
14	2
15	1
16	1
17	4
18	1
19	29
20	14
21	1
22	1
23	14
24	2
25	1
26	2
27	4
28	1
29	2
30	4
31	5
32	12
33	2
34	1
35	2
36	2
37	9
38	16
39	1
40	2
41	80
42	1
43	2
44	4
45	2
46	3
47	16
48	2
49	2
50	2
51	1
52	15
53	960
54	15
55	1
56	4
57	3
58	1
59	14
60	1
61	6
62	20
63	1
64	3
65	946
66	6
67	1
68	18
69	10
70	1
71	4
72	1
73	5
74	42
75	4
76	1
77	828
78	1
79	1
80	2
81	1
82	12
83	2
84	6
85	4
86	30
87	3
88	3022
89	2
90	1
91	1
92	8
93	2
94	4
95	4
96	2
97	11
98	8
99	2
100	1
101	2
102	1
103	56
104	2
105	12
106	1
107	4
108	5
109	15
110	2
111	1
112	1
113	4
114	3
115	2
116	16
117	3
118	1
119	46
120	1
121	2
122	6216
123	865890
124	2
125	16
126	4
127	165
128	72
129	5
130	64
131	14
132	1
133	2
134	50
135	2
136	279
137	12
138	2
139	1
140	2
141	6
142	1
143	4
144	1
145	3
146	4
147	4
148	1
149	2
150	14
151	1
152	8
153	4
154	1
155	6
156	1
157	29
158	1620
159	16
160	5
161	2
162	1
163	1
164	20
165	1
166	5
167	24
168	1
169	1
170	2
171	52
172	99
173	2
174	1
175	15
176	4
177	1
178	8
179	46
180	2484
181	2
182	396
183	22
184	3
185	208
186	3
187	8
188	10
189	1
190	1
191	2
192	15
193	1
194	2
195	1
196	60
197	520
198	4
199	8
200	60
201	2
202	46773
203	2
204	1
205	2
206	2
207	1
208	9
209	2
210	1
211	2
212	4
213	4
214	118
215	2
216	1
217	5
218	44
219	1
220	5
221	24
222	3
223	14
224	2
225	5
226	2
227	16
228	24
229	41
230	2
231	4
232	1
233	16
234	10
235	15
236	4
237	1
238	40
239	50
240	3
241	38
242	4
243	6
244	14
245	4
246	1
247	1
248	604
249	1851
250	3
251	102978
252	11
253	2
254	2
255	4
256	4
257	1344
258	2
259	2
260	2
261	5
262	4
263	10
264	5
265	3
266	2
267	1
268	6
269	1436
270	3
271	12
272	16680
273	1
274	2
275	980
276	3
277	24
278	2
279	1
280	1
281	2
282	276
283	14
284	36
285	2
286	2
287	4
288	1
289	1
290	24
291	1
292	5
293	44
294	1
295	6
296	4
297	14313
298	60670
299	4
300	2
301	2
302	16
303	5
304	70
305	52
306	8
307	203
308	8
309	2
310	1
311	2
312	4
313	4
314	12
315	1
316	5
317	128
318	2
319	564
320	2
321	2
322	17
323	2
324	6
325	2
326	64756
327	24
328	1626
329	480
330	1
331	4
332	112
333	1
334	2
335	4
336	2
337	1
338	272
339	10
340	324
341	84
342
343	1
344	2
345	1
346	4
347	69660
348	2
349	1
350	270
351	5
352	1
353	2
354	21
355	2
356	528
357	3
358	1
359	104
360	4
361	6
362
363	142876
364	3
365	2
366	2
367	5
368	36
369	8
370	1
371	22
372	3
373	2
374	52
375	6
376	8
377	16
378	5
379	1
380	2
381	3
382	1
383	4
384	1
385	9
386	2
387	4
388	41
389	70
390	2
391	3
392	4
393	1
394	3
395	2
396	1
397	44
398	56
399	1
400	2
401	16
402	112
403	1
404	2
405	1
406	1
407	500
408	76
409	3
410	42
411	2
412	9
413	4
414	2
415	1
416	180
417	1
418	9
419	4338
420	10
421	9
422
423	2
424	4
425	2
426	3
427	1
428	2
429	6
430	3
431	76
432	99
433	2
434	882
435	1
436	1
437	108
438	126804
439	15
440	4
441	2
442	16
443	4
444	2
445	3
446	2
447	3
448	1
449	14
450	20
451	2
452	71940
453	6
454	1
455	1168
456	1
457	1
458	2
459	3
460	5
461	26
462	80
463	10
464	36
465	2
466	4
467	40
468	18
469	9
470	24
471	263
472	45
473	72
474	2
475	4
476	60
477	1
478	9
479
480	15
481	5
482	4
483	2
484	2
485	20
486	4
487
488	10
489	2
490	1
491	2
492	27
493	5
494	144
495	2
496	1
497	4012
498	1937
499	20
500	2
501	1
502	3
503	892
504	9
505	2
506	2
507	3
508	4
509	194
510	3
511	15
512
513	1
514	2
515	6
516	6
517	8
518	2
519	1
520	24
521	8
522	105
523	2
524	814
525	2
526	1
527	328
528	2
529	6
530	4
531	1
532	1
533	6
534	1
535	45
536	9762
537	56
538	1
539	2
540	12
541	1
542	156680
543	10041
544	10
545	1822
546	131
547	11
548	16
549	23
550	4
551	248
552	1
553	752
554	22
555	1
556	6
557	60
558	32
559	9
560	2
561	1
562	5
563	6
564	38064
565	2
566	2
567	1
568	1
569	148
570	160
571	10
572	1832
573	1
574	17
575	34
576	3
577	3
578	2764
579	2
580	1
581	44
582	4
583	1
584	2
585	1
586	2
587	916
588	106
589	4
590	26
591	10
592	99
593	1660
594	2
595	11
596	370
597	15
598	4
599	1018
600	5
601	17
602
603	9
604	1
605	6
606	1
607	3
608	2
609	5
610	1
611	274
612	5
613	1
614	30
615	3
616	2
617	16
618	18
619	2
620	972
621	3
622	1
623	2
624	18
625	1
626	8
627	1
628	8
629	2
630	5
631	3
632	4
633	5
634	84822
635	74
636	1
637	11
638	6
639	3
640	9
641	4
642	3
643	1
644	14
645	1
646	5
647	24
648	6
649	34
650	22
651	131
652	71
653	6
654	2
655	142
656	236
657	4
658	786
659	4
660	3
661	237
662	28
663	2
664	60
665	70
666	2
667	3
668	58
669	1
670	2
671	12
672	1
673	24
674	6342
675	9
676	3
677	320
678	1
679	2
680	40
681	2
682	616
683	738
684	3
685	6
686	64
687	1
688	172
689	4
690	3
691	314
692	4
693	5
694	227
695	2
696	4
697	1
698	10
699	23
700	23
701	8662
702	1
703	6
704	12
705	197
706	70
707	12
708	6
709	9
710	2
711	21
712	4
713	14
714	2
715	3
716	8
717	59
718	144
719	16
720	3
721	1
722	972
723	2
724	3
725	4
726	6
727	1
728	16
729	1
730	4
731	14
732	1803
733	5
734	2
735	3
736	1
737	12
738	86
739	12
740	766
741	4
742	11
743	238
744	1
745	71
746	40
747	20
748	1
749	6
750	2
751	870
752	4
753	40
754	2
755	2
756	1
757
758	3014
759	802
760	267
761	4
762	1
763	1
764	2
765	2
766	81
767
768	65
769	1
770	116
771	6
772	3
773	2
774	1
775	3
776	6
777	3
778	4
779	1014
780	5
781	4
782	1200
783	1
784	6
785	2
786	1
787	20
788	13540
789	2
790	5
791	26
792	12
793	1
794	962
795	52
796	9
797	5188
798	5
799	1
800	12
801	53
802	1
803	2
804	3
805	16
806	2
807	36
808	2
809	10
810	2440
811	99
812	44
813	1
814	3
815	2
816	6
817
818	2
819	1
820	1
821	14
822	3
823	1089
824	2
825	5
826	1
827	8452
828	1
829	4
830
831	6
832	65
833	18
834	4
835	24
836	1048
837	1
838	104
839	2172
840	1
841	12
842	52
843	4
844	9
845	10
846	18
847	1
848	2
849	3
850	5
851	12
852	45
853	2
854	2
855	2
856	1
857	12
858	1
859	2
860	4
861	1
862	11
863	4
864	322
865	1
866	14
867	65
868	5
869	12288
870	4
871	3
872
873	2
874	3
875	58
876	1
877	4
878	114
879	1
880	2
881	2
882	3
883	2
884	2
885	4
886	33
887	32
888	5
889	3
890	37376
891	1
892	596
893
894	11
895	1
896	4
897	27
898	1
899	2
900	1
901	23
902	16
903	302
904	3
905	4
906	4
907	327
908	2
909	3
910	2
911	4
912	1
913	36
914	18
915	3
916	1
917	44
918	653
919	1
920	142
921	1
922	41
923	4470
924	17
925	2
926	10
927	96
928	1
929	28
930	4
931	3
932
933	4
934	51
935	84
936	26
937	56
938	12
939	15
940	42
941	24
942	243
943	5
944	4
945	36
946	2
947	120
948	36
949	9
950	5956
951	1
952	41
953	60994
954	2
955	173
956	714
957	17
958	10
959	10
960	1
961	1
962	64
963	81
964	50
965	2
966	6
967	12
968	6
969	2
970	1
971	6
972	15
973	30
974	1622
975	1
976	3
977	48
978	26
979	4
980	2
981	118
982	1
983	2
984	33
985	3
986	4
987	8
988	1
989	68
990	3
991	1
992
993	1
994	1
995	110
996	5
997
998	34
999	3
1000	8
1001	28
1002	1
1003	1
1004	29684
1005
1006	3
1007
1008	1
1009	18
1010	4
1011	2
1012	1
1013	6
1014	2
1015	71
1016	14
1017	1
1018	16
1019	444
1020	3
1021	1
1022	6064
1023	1
1024	38

Williams primes

Main	Williams prime

Type MM: (b-1)•bⁿ-1

Type MP: (b-1)•bⁿ+1

Type PM: (b+1)•bⁿ-1

Type PP: (b+1)•bⁿ+1

@@ Line 1: / Line 1: @@
-{{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b-1)|b|n}}, least ''n''-values}}
+{{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b-1)|b|n}}, least {{Vn}}-values}}
 ==Description==
-Here are shown the least ''n''-value ≥ 1 for any base ''b'' with 2 ≤ ''b'' ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|+|(b-1)|b|n}}.
+Here are shown the least {{Vn}}-value ≥ 1 for any base {{Vb}} with 2 ≤ {{Vb}} ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|+|(b-1)|b|n}}.
 ==History==
-{{HistF|2019-05-29|3014 ([[Williams prime MP 758|b=758]])|Karsten Bonath}}
+<div style="width: 35em; height:20em; overflow:auto;">
-{{HistF|2019-05-28|38064 ([[Williams prime MP 564|b=564]])|Karsten Bonath}}
+{{HistF|2024-08-13|{{NWi|MP|438|126804}}|Predrag Kurtovic}}
-{{HistF|2019-05-25|6342 ([[Williams prime MP 674|b=674]])|Karsten Bonath}}
+{{HistF|2024-06-19|{{NWi|MP|542|156680}}|Lleyton Joseph}}
-{{HistF|2019-05-25|8662 ([[Williams prime MP 701|b=701]])|Karsten Bonath}}
+{{HistF|2024-05-11|{{NWi|MP|123|865890}}|Predrag Kurtovic}}
-{{HistF|2019-05-23|1822 ([[Williams prime MP 545|b=545]])|Karsten Bonath}}
+{{HistF|2019-08-06|{{NWi|MP|634|84822}}|Karsten Bonath}}
-{{HistF|2019-05-22|9762 ([[Williams prime MP 536|b=536]])|Karsten Bonath}}
+{{HistF|2019-06-20|{{NWi|MP|953|60994}}|Karsten Bonath}}
-{{HistF|2019-05-22|4012 ([[Williams prime MP 497|b=497]])|Karsten Bonath}}
+{{HistF|2019-06-15|{{NWi|MP|1004|29684}}|Karsten Bonath}}
-{{HistF|2019-05-13|142876 ([[Williams prime MP 363|b=363]])|Karsten Bonath}}
+{{HistF|2019-06-14|{{NWi|MP|974|1622}}|Karsten Bonath}}
+{{HistF|2019-06-13|{{NWi|MP|923|4470}}|Karsten Bonath}}
+{{HistF|2019-06-11|{{NWi|MP|950|5956}}|Karsten Bonath}}
+{{HistF|2019-06-09|{{NWi|MP|890|37376}}|Karsten Bonath}}
+{{HistF|2019-06-06|{{NWi|MP|869|12288}}|Karsten Bonath}}
+{{HistF|2019-06-06|{{NWi|MP|839|2172}}|Karsten Bonath}}
+{{HistF|2019-06-06|{{NWi|MP|836|1048}}|Karsten Bonath}}
+{{HistF|2019-06-03|{{NWi|MP|827|8452}}|Karsten Bonath}}
+{{HistF|2019-06-02|{{NWi|MP|797|5188}}|Karsten Bonath}}
+{{HistF|2019-06-02|{{NWi|MP|788|13540}}|Karsten Bonath}}
+{{HistF|2019-05-31|{{NWi|MP|782|1200}}|Karsten Bonath}}
+{{HistF|2019-05-30|{{NWi|MP|593|1660}}|Karsten Bonath}}
+{{HistF|2019-05-30|{{NWi|MP|578|2764}}|Karsten Bonath}}
+{{HistF|2019-05-30|{{NWi|MP|572|1832}}|Karsten Bonath}}
+{{HistF|2019-05-29|{{NWi|MP|758|3014}}|Karsten Bonath}}
+{{HistF|2019-05-28|{{NWi|MP|564|38064}}|Karsten Bonath}}
+{{HistF|2019-05-25|{{NWi|MP|674|6342}}|Karsten Bonath}}
+{{HistF|2019-05-25|{{NWi|MP|701|8662}}|Karsten Bonath}}
+{{HistF|2019-05-23|{{NWi|MP|545|1822}}|Karsten Bonath}}
+{{HistF|2019-05-22|{{NWi|MP|536|9762}}|Karsten Bonath}}
+{{HistF|2019-05-22|{{NWi|MP|497|4012}}|Karsten Bonath}}
+{{HistF|2019-05-10|{{NWi|MP|363|142876}}|Karsten Bonath}}
+{{HistF|2018-06-28|{{NWi|MP|452|71940}}|Karsten Bonath}}
+{{HistF|2018-06-13|{{NWi|MP|251|102978}}|Karsten Bonath}}
+{{HistF|2018-05-15|{{NWi|MP|326|64756}}|Karsten Bonath}}
+{{HistF|2018-05-14|{{NWi|MP|347|69660}}|Karsten Bonath}}
+{{HistF|2018-04-27|{{NWi|MP|202|46773}}|Karsten Bonath}}
+{{HistF|2018-04-23|{{NWi|MP|298|60670}}|Karsten Bonath}}
+</div>
-==Wanted values==
+==Wanted values [searched range]==
-Given here bases and known searched ranges:
+<div style="column-count:6;-moz-column-count:6;-webkit-column-count:6">
-*Searched by [[Conjectures 'R Us]]
+{{#dpl:
-**123 [250000]
+|debug=1
-**342 [100000]
+|namespace=
-**438 [100000]
+|category=Williams prime MP without
-**487 [100000]
+|include={Williams prime}:WiBase,{Williams prime}:WiBase,{Williams prime}:WiMaxn
-**757 [100000]
+|mode=userformat
-**997 [100000]
+|format=<ul type="disc">,<li>,</li>,</ul>
-**1005 [100000]
+|secseparators=[[Williams prime MP ,|,,]],&nbsp;[,]
-*Others
+|order=ascending
-**[[Williams prime MP 362|362]] [{{GP|Williams prime MP 362|WiMaxn}}] {{#if:{{GP|Williams prime MP 362|WiReserved}}|<b>RESERVED!</b>}}
+|ordermethod=sortkey
-**[[Williams prime MP 479|479]] [{{GP|Williams prime MP 479|WiMaxn}}] {{#if:{{GP|Williams prime MP 479|WiReserved}}|<b>RESERVED!</b>}}
+|oneresultheader=<b>There is one sequence</b>:
-**[[Williams prime MP 422|422]] [{{GP|Williams prime MP 422|WiMaxn}}] {{#if:{{GP|Williams prime MP 422|WiReserved}}|<b>RESERVED!</b>}}
+|resultsheader=<b>There are %PAGES% sequences</b>:
-**512, 542, 572, 578, 593, 602, 634, 767, 782, 788, 797, 817, 827, 830, 836, 839, 869, 872, 890, 893, 923, 932, 950, 953, 974, 992, 1004, 1007
+|noresultsheader=<b>There are no sequences.</b>
+}}</div>
+'''Notes:'''
+*Base 512: {{NPr|511|n}} (first {{Vn}}-value divisible by 9).
+*Bases 342, 487, 757, 997, 1005 are sometimes reserved and worked on by [[Conjectures 'R Us]].
 ==Data==
@@ Line 35: / Line 67: @@
 <div style="width: 12em; height:50em; overflow:auto;">
 {| class="wikitable sortable" style="height: 200px"
-! Base
+! data-sort-type="number" class="fixhead" | Base
-! ''n''-value {{#for_external_table:<nowiki/>
+! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/>
 {{!}}-
 {{!}} style="text-align:right" {{!}} {{{b}}}
@@ Line 43: / Line 75: @@
 |}
 </div>
-==External links==
+{{Navbox Williams primes}}
-*see [[Williams prime]]
 [[Category:Williams prime MP| ]]

Difference between revisions of "Williams prime MP least"

Latest revision as of 00:47, 8 September 2024

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