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March 4

Prime gap

Has there been a proof that every next successive prime is always less than twice its previous prime (i.e.: $P_{i+1}<2P_{i}$ )? I am not sure if this statement is implemented in the Prime gap article.Almuhammedi (talk) 08:14, 4 March 2025 (UTC)[reply]

This is Bertrand's postulate, which was proved by Chebyshev in 1852. It is mentioned in the section Prime gap § Upper bounds. ‑‑Lambiam 08:25, 4 March 2025 (UTC)[reply]

Thank for reply and I also found it in the article.Almuhammedi (talk) 08:30, 4 March 2025 (UTC)[reply]

Does such number always exist?

Give two integers m>=1, n>=2, is there always a number of the form (m*generalized pentagonal number+1) which is Fermat pseudoprime base n? 220.132.216.52 (talk) 15:39, 4 March 2025 (UTC)[reply]

This is a list for 1<=m<=64 and 1<=n<=64, 0 if the corresponding generalized pentagonal number is larger than the 16777216th generalized pentagonal number (0 is the 0th generalized pentagonal number, 1 is the 1st generalized pentagonal number).

m n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63	64
1	6	15	4	9	6	91	8	9	10	21	12	25	14	15	16	33	18	91	20	21	22	45	24	25	26	27	28	57	30	361	32	33	34	35	36	253	38	39	40	81	42	85	44	45	46	93	48	49	50	51	52	105	54	55	56	57	58	117	60	121	62	63	64	65
2	194221	4371	5461	5461	2603381	4033	645	561	30889	1023121	1387	1105	40604152161	561	219781	561	587861	1387	0	6601	5461	63560685	154101	1194649	241001	3277	2701	3277	993420289	2704801	320454751	2465	1141141	154910869	656601	49141	0	41041	0	4681	0	14491	0	62745	57421	1092547	857762415475351	15841	679729	91001	65077	34945	0	23634181	90751	40827473	2794873567201	448921	0	40622401	2508787938931	41665	1387	41665
3	8401	432821	121	6601	286	91	39275942861	121	1891	121	19724881896721	1105	0	2465	1891	13183873	2187114805	91	286	1541	29020321	1541	15457	121	5551	12403	703	9549541	379029421	16531	0	2465	2480017	65485	23521	49141	5127497371	41041	17076979798165	8401	0	512191	614514345901	1541	91	1133441	0	15841	96139	428108801	455533	7846385	156748879	1891	90751	30857	286	30857	0	121	0	0	17216641	0
4	15051	15	5461	5461	53131	91	15	561	1891	1023121	1387	85	60873151	15	1891	561	33355	91	6736901215801	6601	5461	1695	154101	1194649	5551	3277	703	3277	993420289	451	320454751	2465	227767	154910869	656601	49141	54058181295301	41041	42121	4681	0	85	0	28645	91	1092547	8399371	15841	679729	14351	65077	34945	0	1891	90751	40827473	129558009211	227767	0	40622401	2508787938931	435	1387	41665
5	7813	24211	4	5461	2603381	15751	303268552133	561	1891	1576261	136137	7813	6184050601	561	1891	561	5611	217	7449	1541	5461	1541	57025874137	431434441	265651	5461	1891	1141141	140966101	781	110972716	11041	1141141	341531	656601	30673	0	9881	1105118600052481	9881	9809069344817	178482151	102310037245681	1541	21789901	0	0	15841	341531	68251	939727	781	0	1891	35935711361551	3270933121	0	0	0	15562561	100651	7813	2457244165321	27700609
6	178482151	185	301	6601	1111	1261	732733	94697	97921	1261	407264221	481	16589	2465	36301	38081	35	217	1628949421	301	16322041	36301	9227417	4377277921	301	7374121	2701	167692141	993420289	1380751	0	481	386649121	35	403287950101	1261	4603397328001	9881	3589	481	0	178482151	2786057	340561	181351	3279704502724651	0	15841	2704801	1370778751	84151	1992641	0	392099401	90751	171361	363091	4658827345201	0	301	0	63733645	173377	7195975489
7	6	25	14794081	1073477505	6	703	0	561	97921	2101	78937	25	7164662961	561	88831	561	350065	10621	0	19529401	2101	74563831	78937	25	118301	3277	325	3277	993420289	2101	0	1825	114841	1282947009884051	656601	270990721	0	3326759288281	0	64681	0	512191	0	62745	38357866	0	39079399527901	7519441	29891	571389001	84151	737618701	39623838801	392099401	35935711361551	126673	14822750251	227767	0	2101	0	23683666751	20356713256321	1123201
8	194221	45	5461	9	42001	133	645	9	30889	21	133	481	16589	561	219781	561	587861	1387	68191761	21	5461	45	154101	1194649	651	3277	2701	3277	993420289	2704801	63	65	1141141	511	96321	49141	0	7107	0	481	142681	14491	675928828074501	45	57421	231	848715305621	15841	76049	91001	65077	105	0	23634181	24311	13833	30571087933	117	19953801	5090821	59318841	63	1387	65
9	8	8695	4	205	286	91	8	121	1891	121	1288	1105	94102707427	2465	1891	13183873	205	91	286	1541	429976	1541	15457	121	2501	12403	28	1141141	379029421	16531	0	2465	1141141	511	23521	49141	5127497371	41041	42121	8401	616	512191	614514345901	1541	91	1092547	10639657666	15841	96139	91001	52	7846385	156748879	1891	2806	30857	286	697	0	121	0	0	946	27700609
10	8401	7758601	5461	9	23661	91	1233	9	512461	460251	5943301	481	23661	99	399001	33	0	91	6580849460329	6601	5461	760761	44039315321	237169	118301	5461	703	1141141	993420289	451	320454751	33	1141141	2585701	656601	1884961	0	7107	0	481	3119201177501	178482151	1449006218080591	340561	91	69921	0	15841	99	328601001	43205910721	47287045505701	119436866341	1583821	537368261	1233	0	51292417	8772121	40622401	145181	520801	607321	21553729
11	38963	15	41329	6601	496479061	133	15	31417	10	7207201	133	481	4880464780621	15	88831	33153	0	8911	0	6601	127664461	17711	5541015791	5041	346801	4577	190	1131929	993420289	1638781	646590205361	481	2975281	384541	134204071	793	0	993441390811	119341	481	0	178482151	2786057	14521	57421	15101893	0	15841	5444489	171601	455533	34945	0	3180871	9602561	30857	29329786431331	30857	97351	40622401	0	0	14019391	347229121
12	377	75927853	339901	6601	35881	91	105635531	110209	1891	3157921	133	145	11077	2465	1891	3553	1307944841608441	91	73645	6601	24016231	6335671	533708101	7252249	5551	12403	703	167692141	993420289	34861	0	65	52192141	341531	134204071	49141	0	41041	37598854021	2041	3652809721	512191	0	340561	91	0	15794788511881	15841	76049	103601	2041	3533245	66037442017	1891	90751	126673	295945	0	97351	40321	0	185012166588481	22766689	65
13	6	663171	4	21	6	20881	85	561	1891	21	12	85	630631	561	1891	561	0	8911	6580849460329	21	19817071	74563831	8553566770201	744769	49051	12403	1891	167692141	122961	26281	2817010891171	2465	22177	8841	23521	49141	0	7107	21431239831	87676681	0	85	0	5149	68401	231	0	15841	2704801	15051	0	105	0	1891	276	4966921	0	0	0	23521	0	964411	184339	27700609
14	178482151	15	781	841	1111	2257	15	561	1891	1625261	1660521721	481	40604152161	15	1891	561	587861	12871	39	1541	841	1541	15457	841	6501	3277	1891	3277	993420289	781	3310801	65	1141141	0	23521	793	86822750785	39	12871	481	0	11210431	0	1541	68401	7689683	1625261	3650401	1885521	571389001	455533	781	2535819511288141	1891	3400585509751	841	7211832433	705088021	0	23521	0	0	529201	65
15	38963	432821	742	6601	7710830881	50401	0	15409	1891	3035251	407264221	2091013	14	8650951	1891	54913	0	8911	6580849460329	1541	24016231	1541	111428377	848615161	438751	3277	946	3277	993420289	2704801	2817010891171	15841	227767	0	368971	382537	54058181295301	41041	12871	87676681	0	178482151	0	1541	1639171	1092547	0	1921	1207361	91001	60691	1992641	0	1891	323487451	3270933121	0	227767	0	8701	0	964411	946	250958401
16	15051	15	5461	5461	53131	91	15	561	1891	51	1387	85	60873151	15	1891	561	33355	91	495331	6601	5461	1695	54741	1194649	51	3277	595	3277	3655	451	320454751	2465	227767	2585701	656601	1261	54058181295301	41041	42121	4681	0	85	0	28645	91	54741	1625261	15841	679729	51	65077	34945	0	1891	90751	40827473	14822750251	227767	0	19196101	100651	435	1387	41665
17	8	45	4	9	1111	91	8	9	21781	261	33422278670551	145	48936889	9069229	16	38081	350065	91	3991	6601	29020321	45	81696277	2117932441	550551	12403	8911	1141141	993420289	781	70235944639183	4033	1141141	48799900801	656601	77293	4603397328001	41041	2297296	321201	0	1711711	0	45	91	0	729081673	15841	1003276	387001	455533	261	39623838801	5767201	3563561	13833	3991	484897459	0	11463980221	0	0	60505201	23522415840001
18	13021	25	270481	49	2603381	133	85	493697	30889	221	133	25	25039	2465	399001	344641	11425	1387	0	6601	29020321	171865	44039315321	25	4864501	4577	325	1141141	993420289	451	27100010653	65	227767	11221	29821	28153	72099263825	41041	119341	20803481	14439537113	85	0	221	21789901	323	1063705	49	76049	68251	60691	3533245	637	18631	90751	126673	285445824931	227767	1238404685281	11221	1634786208817441	931	1387	65
19	6	15	5461	9	6	343	15	9	1891	974101	65199443	1105	16589	15	1891	561	18	4681	871151	6601	5461	45	795341	49	15251	5461	1891	167692141	993420289	16531	158315761	2465	4906441	0	656601	49141	21807061	41041	12871	4681	48346461652321	5739511	1513171	45	139935601	1241819958093001	1035	49	2281049	91001	3304393	34945	637	1891	2798181	13833	363091	5060311179841	0	66421	0	3108805	160730389	27700609
20	69231	54811	5461	21	2603381	133	0	57	1891	21	133	5072113	47139	561	1891	561	0	8911	159601	21	5461	1758373	8553566770201	78961	2501	5461	1891	57	379029421	2761	320454751	1281	114841	5271	656601	382537	0	41041	8219251	64681	0	859951	0	340561	21789901	231	1625261	15841	859951	68251	455533	1992641	8984137	1891	52979415061	57	24653085537061	28045083749761	0	5090821	1561336237	0	267121	1123201
21	178482151	8695	4	194041	2603381	703	85	31417	10	221	33422278670551	85	5938141	2465	88831	234433	74986661	3781	20	11135555341	22681	8678935	1433407	0	265651	68251	55	167692141	137180905	3781	849556	65	386649121	0	327818821	793	0	202236517561	53902799101009	7611281	0	85	0	221	23910076	0	0	164737	1207361	68251	27133	0	930151	55	35935711361551	553393	0	62641	451556029	2975701	21331213	0	42171555830041	65
22	15051	432821	3241	21	53131	91	645	31417	1891	21	19724881896721	1105	630631	169	1891	33153	32743	91	19454275691101	21	25448893	485	7263217	169	5551	68251	1891	167692141	993420289	28310101	60474549583	161	2453415361	69	96699821	1261	638747426521	3801	436567070941	2041	3119201177501	178482151	0	1541	91	15101893	51018930145	2737	2281049	15051	2041	105	0	1891	2118006001	216431041	3517585	247638376441	0	40622401	0	0	292667761	7229815361
23	15051	1065	22	2519819281	2603381	91	285286	561	316711	13351	9279489781	265	5938141	169	399001	33	1153578770241	91	1749	14400541	22	265	268181	169	2501	506891841	946	311151821	993420289	451	0	33	22177	0	656601	828073	638747426521	41041	119341	481	0	11210431	28897	5149	91	553	0	15841	76049	15051	30058381	184302384553	0	31861	35935711361551	3270933121	1902433	2321	0	4166701	0	20461	946	1123201
24	7813	25	1771	6601	433457811346561	553	777798580483	11972017	512461	63701	407264221	25	31691401	2465	399001	94753	350065	3781	0	1541	6931	1541	7263217	25	15251	169027	325	167692141	993420289	3781	1518924044079001	1825	6931	88205654698141	63701	4537	140868991	41041	42121	87676681	0	178482151	434288821	1541	181351	553	0	2737	2385811	0	455533	0	0	3781	6931	3270933121	115	533187724527601	0	5090821	554337492547021	7813	3816279181	1123201
25	6	24211	4	5461	6	91	8	561	1891	5321	12	7813	66	561	1891	561	868	91	39	1541	5461	1541	24	431434441	2926	5461	28	1141141	44081	451	849556	11041	232	341531	8646	30673	4153554082651	39	4564	9881	616	512191	81608626	1541	91	1092547	1625261	15841	341531	15051	939727	781	0	1891	276	3270933121	15097194421	0	97351	15562561	100651	7813	946	27700609
26	27	15	701521	9	35881	133	15	9	3385	333761	133	25	27	15	88831	561	0	217	0	6601	341797	45	0	25	655051	27	703	137257	2083129	67861	493703331521	225	227767	384541	656601	49141	75	4590913687801	164275862401	5041	0	925	176441982624931	45	169396921	15101893	1035	15841	589	68251	5167151043121	0	0	3180871	35935711361551	13833	17914028401	227767	0	5090821	0	612685	487621	1123201
27	8401	432821	121	5461	26	91	29790368791	121	1891	121	133	481	16589	365	1891	2993	2993	91	286	1541	5461	1541	15457	121	26	5461	703	9549541	379029421	5611	201180401657	65	2480017	65485	23521	49141	5127497371	9881	8219251	481	74621	512191	614514345901	1541	91	1133441	64950241	15841	29891	32551	455533	365	4572841	1891	90751	30857	286	30857	0	121	3192646343041	0	17216641	65
28	27	45	529	9	433457811346561	703	27133	9	361	261	2104873	145	27	561	88831	561	587861	205147	19454275691101	7706401	265651	45	0	361	265651	27	703	3277	993420289	361	0	2465	227767	341531	656601	289081	0	18314101	29070037629361	2041	2871	512191	87	45	2289421	73974930621373	857762415475351	7519441	96139	32551	2041	261	0	5767201	1431	13833	295945	227767	0	40622401	0	0	583637041	27700609
29	52	15	4	21	2603381	91	15	561	97921	21	136137	481	14	15	219781	561	35	91	9305101345	21	4411	1495628245	10477460941	0	651	68251	28	1141141	993420289	5611	70235944639183	481	227767	35	4411	793	14615371	41041	469	481	0	178482151	489124488346957	5149	91	231	857762415475351	15841	679729	68251	52	105	156748879	361801	0	708401	0	871	0	5409601	0	520801	4411	2213121
30	8401	6031	39061	49	39061	91	27133	6817	901	7207201	133	50881	8142831789821	8650951	117181	38081	9696209251	91	0	6601	29020321	12913590691	142930283186881	49	27001	3277	703	3277	993420289	55651	2817010891171	11041	7561159	170045313273221	46349511957001	3158101	0	9881	469	8401	493	178482151	0	15181	91	15181	0	49	589	91001	27133	2205841	637	5767201	385887787445161	126673	9490501	871	9823501	901	0	931	418491457021	250958401
31	6	15	17767	49	6	133	15	561	10	1023121	133	481	66	15	88831	561	0	8911	7449	6601	514753	676243261	9585058411	49	346801	12403	946	311151821	30	451	0	65	22177	16219	656601	270990721	638747426521	7107	42121	481	0	178482151	14191	62745	2804491	225998907120001	22004248501	49	679729	68251	804471553	1525161	637	392099401	196972051	126673	0	413261067677	0	40622401	0	931	946	65
32	93	25	5461	205	73261	1057	645	561	30889	1023121	1387	25	37360389	561	219781	33	205	217	73645	6601	5461	1542025	9017	25	241001	3277	325	3277	993420289	2761	320454751	33	22177	154910869	656601	793	19685	41041	3068830129	4681	74621	14491	3177529377	62745	4141	93	857762415475351	1057	679729	91001	65077	1353	0	11989	90751	19609801	1067560552045	697	482866385	40622401	2508787938931	41665	1387	41665
33	8	75927853	4	5461	0	703	8	975841	30889	7207201	1660521721	85	16589	2465	16	21361	0	4681	21601483345	6601	5461	1035805	1433407	54289	2501	5461	703	1141141	993420289	67861	32	2465	496	39959521	134204071	49141	86822750785	0	2297296	4681	0	85	3785729117	340561	556921	828911670186601	857762415475351	1057	29891	91001	27133	2722721	0	392099401	35935711361551	30857	975841	30857	0	208801	42787963	1276705	262972711	27700609
34	15051	15	507529	21	2582801	703	15	31417	1891	21	407264221	1362901	5938141	15	1891	33	35	8911	73645	21	29020321	3086623	0	0	1001	20450501527	55	1141141	993420289	145351	0	33	22177	35	482161	382537	445	77	777303541	87676681	0	512191	0	48709	71559635401	231	0	15841	2704801	15051	60691	105	0	55	35935711361551	3270933121	363091	140361	0	5090821	890602441	435	529201	65
35	15051	4371	49771	9	19763205001	1261	4589504641	9	30889	51	305039653073	78013	16589	561	20026	561	74986661	1261	1628949421	3075201	19951	20068621	1433407	2601769	51	12403	2701	1141141	993420289	1638781	40653856321	11041	34	22024701289	656601	1261	0	0	375961	321201	2871	178482151	0	340561	9534421	7689683	8399371	1921	266071	51	455533	862681	0	16390351	35935711361551	13833	363091	0	451556029	17121721	705169377232321	0	442	2213121
36	178482151	185	301	6601	1111	91	102565	94697	1891	1261	1387	481	16589	2465	1891	38081	35	91	1628949421	301	17893	36301	9227417	4377277921	301	12403	703	1141141	993420289	5611	428221164823	481	210211	35	134204071	1261	417339784163	9881	3589	481	0	35785	2786057	340561	91	1092547	0	15841	29891	91001	84151	246065	0	1891	90751	171361	363091	135253062127	112546128432151	301	0	63733645	1387	27700609
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--220.132.216.52 (talk) 21:54, 5 March 2025 (UTC)[reply]

Note that (assuming Bunyakovsky conjecture is true), (m*generalized pentagonal number+1) is always composite if and only if m is in {24, 25, 27, 32, 49}, and if m is not in {24, 25, 27, 32, 49}, then there are infinitely many primes of the form (m*generalized pentagonal number+1), see Wikipedia:Reference desk/Archives/Mathematics/2023 May 20#For which natural numbers n.2C this sequence contains infinitely many primes.3F 220.132.216.52 (talk) 12:42, 10 March 2025 (UTC)[reply]

For m in {24, 25, 27, 32, 49}, there are no primes of the form (m*generalized pentagonal number+1) because:

24 * generalized pentagonal numbers + 1 are always squares
25 * generalized pentagonal numbers + 1 are always generalized pentagonal numbers
27 * generalized pentagonal numbers + 1 are always triangular numbers
32 * generalized pentagonal numbers + 1 are always generalized octagonal numbers
49 * generalized pentagonal numbers + 1 are always in OEIS: A144065 (generalized pentagonal numbers-1, n*(n+5)/6 with (positive or negative or 0) integer n)

And all generalized polygonal numbers (other than the trivial cases, i.e. the generalized pentagonal numbers 1, 2, 5, 7, the triangular numbers 1, 3, the generalized octagonal numbers 1, 5) are composite. (and the only prime in OEIS: A144065 is 11) 220.132.216.52 (talk) 12:53, 10 March 2025 (UTC)[reply]