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 2021-03-03, 13:59 #1 robert44444uk     Jun 2003 Oxford, UK 36148 Posts Multithreading existing code I am wondering if there are c programmers out there who could take a great piece of prime finding software and make it multi thread. The software helps to find constant k such that the power series k*2*n+/-1, variable n, is very prime. The existing software was written by Robert Gerbicz and can be found at: https://sites.google.com/site/robertgerbicz/payam payam2.c
2021-03-03, 16:57   #2
masser

Jul 2003

2·5·157 Posts

Quote:
 Originally Posted by robert44444uk The software helps to find constant k such that the power series k*2*n+/-1, variable n, is very prime.
Did you mean k*2^n+/-1?

I have questions about the definition here.

Is there a scientific definition for "very prime"? We know there is a set of k values for which the series, k*2^n+1, is never prime. Do we know that a k value (call it k_many) that produces many primes for say, n < 1000, will have have a higher probability of generating a prime for a large value of n = N (say N=10^6) than a k value (call it k_few) that produced few primes for n < 1000?

Assume we trial factor both k_many*2^N+1 and k_few*2^N+1 up to F = 2^B, where B is some value greater than 50. Both terms "survive" trial factoring, which means no factors were found. [Maybe we should also assume that F>>max(k_many,k_few)] Does the k_many term have a greater probability of being prime? If so, that is what I would call a "very prime" sequence.

My larger question is, does the software find k values that generate "dense-after-trial-factoring" sequences or "very prime" sequences?

Last fiddled with by masser on 2021-03-03 at 17:00

2021-03-03, 17:30   #3
Dr Sardonicus

Feb 2017
Nowhere

4,457 Posts

Quote:
 Originally Posted by masser Did you mean k*2^n+/-1? I have questions about the definition here.
Yes, it was a typo. As the term "power series" indicates, it's k*2^n +/- 1.

From the link in the OP,
Quote:
 By sieving fast code in gmp to find good payam numbers, where k*2^n+-1 is prime for at least 100 n values (1<=n<=10000).
See Payam Number

 2021-03-03, 18:28 #4 robert44444uk     Jun 2003 Oxford, UK 22×3×7×23 Posts Thank you for clarifying Dr S. Very prime sequences arise from "very prime numbers" also been referred to on this site as a VPN. All VPNs are a multiple of k (an integer) and M(p) where M(p) is the product of primes with primitive root 2 less than or equal to p see OEIS http://oeis.org/A001122 for a list of those primes. k are found through application of the Chinese Remainder Theorem. There was an active search for VPN on Mersenneforum which stopped in 2014 https://www.mersenneforum.org/showthread.php?t=9755 The definitions of k etc in the first post of that thread differ to those in this post. But the sentiment is the same. I was thinking with today's computer power, this maybe worth looking at again. Last fiddled with by robert44444uk on 2021-03-03 at 18:29
2021-03-04, 10:37   #5
robert44444uk

Jun 2003
Oxford, UK

22×3×7×23 Posts

Quote:
 Originally Posted by masser My larger question is, does the software find k values that generate "dense-after-trial-factoring" sequences or "very prime" sequences?
Hi masser, the software finds k that provide very prime density. It does more than calculate Nash. It incorporates pfgw and produces output that looks like:

Code:
R 267710937687553 52 100/8531 100/10000 K=620472594867229918205535 iteration=93 I=29389 Sun Oct 19 12:37:40 2014
R 267494780038573 52 100/7065 106/10000 K=619971607128284591472435 iteration=93 I=31889 Sun Oct 19 12:52:54 2014
R 269332638364685 52 100/9216 105/10000 K=624231204193877610405075 iteration=93 I=40088 Sun Oct 19 13:41:49 2014
So K = 619971607128284591472435 in K*2^n-1 produces 100 primes by n = 7065 and 106 primes by n=10000.

K= 267494780038573*M(53) where M(53) = 3*5*11*13*19*29*37*53

Last fiddled with by robert44444uk on 2021-03-04 at 10:40

 2021-03-04, 13:39 #6 kar_bon     Mar 2006 Germany 288110 Posts On my old Prime Database there's still a page with Payam numbers of the Riesel side.
2021-03-04, 14:32   #7
robert44444uk

Jun 2003
Oxford, UK

22×3×7×23 Posts

Quote:
 Originally Posted by kar_bon On my old Prime Database there's still a page with Payam numbers of the Riesel side.
Nostalgia indeed!

Here is (I think) a table of the most recent Riesel records at each level. First column is the cumulative number of primes, and the other columns represent the smallest n value where that number of primes has been found.

Code:
	Best	m28	m36	m52	m58	m60	m66	m82	m100	m106	m130	m138	m148	m162	m172	m178	m180	m196	m210	m226	m268	m292	m316

1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	7
2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	7	11
3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	8	11	17
4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	15	19	22
5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	19	20	27
6	6	6	6	6	6	6	6	6	6	6	6	6	6	6	6	6	6	7	7	9	24	26	47
7	7	7	7	7	7	7	7	7	7	7	7	7	7	7	7	7	8	8	11	11	39	34	124
8	8	8	8	8	8	8	8	8	8	8	8	8	9	9	9	9	10	11	13	17	40	49	135
9	9	9	9	9	9	9	9	9	9	9	9	10	10	12	11	12	11	15	16	19	59	56	176
10	10	10	10	10	10	10	10	10	10	11	11	12	12	15	16	14	17	17	19	27	74	57	200
11	11	11	11	11	11	11	12	11	12	12	13	13	14	18	17	17	20	22	23	38	78	64	231
12	12	12	12	12	13	14	13	12	14	15	14	17	19	21	22	22	24	26	25	46	91	65	273
13	13	14	14	13	15	15	15	16	18	17	18	19	22	23	25	26	27	28	34	47	98	72	276
14	16	16	17	16	18	17	18	19	19	20	22	23	24	24	30	27	30	39	37	53	110	77	284
15	17	17	18	20	20	20	20	20	21	22	27	27	30	32	32	38	37	45	44	60	134	79	300
16	19	19	20	21	22	23	22	22	25	25	31	33	33	38	35	39	41	48	46	72	135	85	320
17	23	23	24	24	25	27	23	26	31	29	36	35	36	39	36	40	44	56	47	77	183	88	419
18	26	26	27	27	29	28	28	29	34	33	40	38	41	40	49	49	46	65	55	83	192	100	444
19	28	28	29	29	31	33	33	33	38	34	46	44	45	53	54	59	54	73	71	94	222	115	465
20	30	30	32	32	34	34	36	36	39	41	49	51	52	55	55	65	61	76	78	105	237	136	483
21	34	34	37	35	38	37	40	37	47	47	53	54	61	67	68	68	67	82	90	115	241	144	490
22	35	35	39	41	43	44	41	42	48	52	56	57	66	74	70	84	77	96	99	119	250	203	506
23	41	43	41	47	48	50	45	43	56	57	58	58	73	81	78	90	89	98	114	133	271	229	651
24	45	45	46	48	53	54	50	54	60	64	72	78	76	88	81	94	100	100	122	151	315	280	653
25	49	51	49	56	57	57	55	56	63	70	81	79	81	94	83	100	104	101	130	153	320	324	670
26	55	55	57	60	59	59	59	62	70	74	85	86	86	106	102	108	115	114	150	185	389	413	710
27	58	61	58	65	63	63	62	64	76	76	86	89	92	114	117	116	138	136	153	188	405	464	739
28	64	64	67	68	73	73	69	80	84	87	87	91	102	122	122	120	141	145	170	193	410	489	804
29	68	68	70	75	77	83	72	83	89	90	93	103	105	129	142	133	145	152	178	216	418	497	874
30	74	74	75	80	83	85	76	92	94	92	94	112	114	137	149	147	146	169	201	225	460	606	995
31	79	79	82	87	88	88	83	102	107	99	113	120	126	140	156	148	159	184	233	265	497	631	1007
32	89	90	92	89	94	94	92	104	110	103	123	140	127	151	164	168	163	192	239	272	500	718	1068
33	91	94	94	91	97	97	100	106	118	109	130	149	146	163	180	175	168	203	244	280	520	724	1086
34	98	98	98	105	110	112	111	115	121	129	136	160	168	173	181	176	171	226	245	292	538	756	1192
35	101	101	101	109	116	122	124	123	146	130	152	167	179	196	201	212	229	245	289	351	607	802	1200
36	115	115	115	118	120	128	130	125	155	136	158	175	180	203	214	231	245	252	303	364	620	837	1333
37	118	118	121	124	124	135	135	159	161	139	160	176	181	224	217	251	260	269	316	393	629	1144	1339
38	125	125	131	129	136	143	147	163	164	142	175	187	220	229	232	259	276	304	330	402	649	1171	1454
39	131	131	146	140	141	153	161	169	169	150	184	207	227	255	254	286	289	329	338	454	689	1290	1466
40	137	137	152	152	146	154	162	171	172	160	196	210	232	263	281	310	290	337	354	489	798	1366	1526
41	141	141	158	162	160	160	183	191	173	169	208	219	250	278	285	321	296	386	385	502	818	1399	1532
42	163	163	163	173	167	167	187	200	179	185	215	226	267	290	302	325	302	407	419	529	828	1439	1850
43	171	171	172	187	180	180	205	224	182	194	234	265	271	316	333	363	357	430	445	546	929	1690	1939
44	176	176	183	193	182	182	223	230	202	220	240	272	274	334	360	383	373	447	502	547	982	1831	2057
45	188	198	213	213	188	188	228	240	237	235	264	296	277	339	366	394	376	477	553	608	1147	1839	2293
46	204	207	217	223	204	204	233	248	239	258	278	304	282	361	378	422	387	478	562	637	1160	1904	2370
47	218	218	226	230	249	249	237	249	243	291	329	316	323	382	422	439	447	480	585	653	1227	2151	2505
48	241	250	257	241	276	262	241	259	248	318	361	341	332	387	432	462	474	534	598	663	1231	2291	2509
49	251	265	261	259	278	275	269	289	251	327	375	351	351	411	466	476	502	571	605	694	1267	2323	2617
50	262	277	270	262	295	283	283	301	322	350	384	359	367	422	484	478	525	582	609	698	1286	2386	3226
51	273	290	273	287	304	290	321	329	350	352	397	407	445	431	504	495	533	646	612	759	1313	2425	3452
52	278	322	278	295	315	317	356	354	361	373	407	425	449	450	560	555	557	667	615	823	1330	2533	3837
53	296	328	296	309	318	348	362	384	392	389	410	452	463	454	587	570	611	710	647	854	1464	2535	3852
54	317	333	327	317	338	374	378	426	397	427	440	456	468	467	588	577	619	723	702	1000	1508	2567	3935
55	328	360	352	328	347	383	384	450	421	445	473	510	533	476	624	579	636	766	782	1037	1547	2581	4016
56	348	367	362	348	352	396	385	456	444	450	495	520	578	490	644	585	674	802	805	1070	1653	2847	4329
57	361	440	407	361	375	403	386	499	446	493	521	541	642	651	677	589	764	850	820	1118	1655	2869	4706
58	371	458	417	371	397	410	416	507	451	517	539	585	665	706	700	590	771	924	938	1220	1711	3260	5472
59	395	461	448	395	404	428	447	528	479	543	591	627	706	711	711	678	826	949	1023	1288	1820	3266	5601
60	411	492	457	462	411	506	455	579	491	551	626	628	763	712	712	725	897	998	1057	1369	1965	3375	5946
61	463	512	471	492	463	556	473	587	544	566	630	664	786	721	721	872	928	1086	1124	1402	2553	3431	6619
62	495	579	499	516	495	581	521	608	548	635	645	713	838	796	796	936	940	1087	1135	1416	2618	3586	7431
63	510	642	510	539	536	601	538	625	550	648	659	795	907	804	804	1004	962	1123	1148	1544	2718	3618	8478
64	519	657	519	571	541	615	563	634	553	693	706	825	934	809	809	1017	977	1133	1219	1545	2801	3698	8657
65	542	698	630	579	542	651	578	651	571	721	791	875	938	824	824	1028	1070	1139	1327	1621	3172	4360	9229
66	564	730	705	599	564	683	595	661	615	737	838	938	1048	884	884	1083	1101	1158	1328	1663	3203	4434	9970
67	565	732	722	628	565	689	615	716	616	794	908	1005	1059	893	893	1090	1127	1287	1348	1711	3329	4771	10372
68	581	745	777	634	581	758	628	755	690	840	950	1057	1089	967	967	1194	1134	1322	1403	1766	3439	5035	10674
69	637	783	795	650	676	772	637	806	719	849	975	1100	1128	1035	1035	1221	1276	1372	1431	1875	3537	5188	10930
70	682	893	864	721	692	789	682	813	749	910	1049	1119	1183	1096	1096	1250	1322	1550	1491	1895	3657	5223	11045
71	695	897	916	732	788	794	695	846	808	972	1156	1156	1289	1184	1184	1442	1382	1648	1518	1942	3919	5552	11734
72	717	967	987	736	832	836	717	867	839	1011	1216	1199	1312	1309	1309	1553	1400	1721	1723	2126	4229	5650	12053
73	720	1113	1065	754	834	863	720	951	867	1023	1221	1271	1422	1455	1635	1584	1450	1748	1796	2305	4295	6071	12833
74	764	1141	1121	764	851	945	783	954	880	1103	1348	1322	1478	1475	1778	1626	1507	1865	1977	2408	4914	6129	13756
75	813	1179	1165	851	938	1001	813	1060	1108	1108	1430	1346	1483	1562	1843	1748	1660	1913	2003	2551	4925	6380	13873
76	828	1284	1233	1046	957	1040	828	1066	1169	1157	1502	1348	1526	1597	1883	1852	1808	1955	2132	2697	5343	7357	13901
77	848	1382	1268	1056	976	1046	848	1127	1172	1239	1508	1356	1691	1670	1912	1964	1878	1975	2238	2965	5648	8159	13921
78	1032	1495	1318	1065	1032	1102	1084	1143	1175	1270	1600	1370	1744	1819	1950	2012	2050	2111	2305	3204	5887	9047	14867
79	1036	1541	1462	1117	1036	1130	1090	1245	1180	1409	1657	1417	1814	1863	2021	2064	2123	2132	2317	3344	6138	9358	16378
80	1040	1613	1501	1204	1040	1293	1175	1247	1190	1521	1702	1573	1821	1876	2092	2294	2216	2449	2367	3608	6362	9834	17123
81	1173	1707	1509	1210	1173	1319	1187	1264	1292	1538	1876	1846	1842	1915	2345	2425	2244	2617	2370	4050	6370	9928	17417
82	1180	1721	1711	1287	1180	1427	1212	1487	1360	1674	1889	1884	1864	1950	2416	2627	2391	2881	2588	4120	6395	9950	18578
83	1302	1757	1818	1382	1372	1465	1302	1493	1416	1698	2058	1887	2148	2321	2480	2639	2554	2921	2671	4243	6939	10529	20832
84	1303	2154	1879	1400	1483	1595	1303	1561	1459	1835	2158	2076	2281	2454	2524	2839	2716	3090	3107	4387	7615	10643	25571
85	1413	2248	1907	1413	1507	1633	1609	1723	1472	1898	2255	2158	2323	2547	2567	3062	2776	3258	3314	4426	7674	12082	27141
86	1421	2276	1987	1421	1628	1662	1784	1762	1525	1998	2360	2309	2439	2655	2709	3225	2813	3350	3322	4563	7741	13201	28935
87	1495	2284	1997	1495	1666	1677	1794	1844	1590	2099	2471	2333	2571	3162	2855	3397	2923	3633	4005	5118	8219	13220	30331
88	1659	2319	2093	1659	1713	1706	1933	1940	1680	2159	2512	2533	2713	3211	3200	3518	2928	3753	4955	5310	8732	13787	30777
89	1721	2353	2098	1884	1815	1815	2003	1945	1721	2191	2550	2634	2787	3236	3374	3797	3410	3771	5085	5398	8935	15024	30820
90	1740	2584	2369	2056	1818	1829	2144	1983	1740	2370	2726	2746	2861	3362	3474	3869	3454	4212	5212	5562	9716	15509	31838
91	1857	2819	2548	2186	1857	2045	2158	2020	1879	2464	2825	2812	3118	3392	3479	4025	3625	4480	5266	5720	10235	16290	33178
92	1917	3030	2658	2233	1930	2108	2170	2024	1917	2617	2921	3074	3258	3747	3591	4193	3649	4560	5391	5842	10291	18304	34444
93	1921	3145	2765	2342	2022	2210	2218	2295	1921	2669	2947	3185	3433	3839	3678	4351	3701	4633	5460	6090	10758	18777	38407
94	1931	3270	3096	2548	2315	2332	2460	2653	1931	2754	3113	3293	3570	4205	3708	4517	3730	4719	5524	7498	11114	19863	44354
95	2081	3583	3458	2615	2529	2378	2635	2744	2081	2851	3248	3349	3831	4314	4075	4849	4268	5494	5589	7629	11289	21699	44491
96	2102	3612	3812	2694	2622	2701	2701	3233	2102	3256	3461	3746	4128	4514	4595	5116	4339	5666	5793	7979	11926	22361	44744
97	2152	3680	3832	2912	2792	2792	2792	3324	2152	3354	3746	4205	4240	5164	4607	5178	4661	5901	6349	8292	12968	22617	46873
98	2185	3834	4282	3299	3228	2929	3839	3548	2185	3791	3877	4261	4656	5504	5016	5207	4926	6421	6360	9129	13676	27204	50741
99	2501	4203	4602	3503	3245	3076	4094	3619	2501	3902	4047	4375	4993	5627	5054	5309	4986	6517	6836	9171	16283	27206	51857
100	2570	4644	5162	3556	3810	3154	4114	3681	2570	3905	4275	4608	5255	5830	5431	5832	5295	7064	6885	9845	20603	27717	62843
101	3045	5840	5241	3943	3945	3158	4476	4063	3045	3950	4567	5134	5412	6052	5609	6249	5753	7601	7483	9915	23009	28071	62934
102	3222	5913	5557	4395	4158	3222	4792	4115	3280	4157	4847	5320	5498	6304	5789	6490	6028	7767	7536	10041	23206	29662	66755
103	3363	5983	5909	4481	4292	3461	5119	4860	3363	4293	5590	5395	5690	6682	6600	6514	6712	8012	7656	10274	27572	34554	68490
104	3429	6008	6480	4559	4299	3481	5361	5134	3429	4501	5886	5448	6263	6825	6765	6561	6934	8757	8532	10720	27752	34780	71484
105	3588	6975	6706	4976	4381	4023	5486	5437	3588	4689	6028	5459	6862	7902	7112	7197	8076	8979	13841	11218	28985	36653	76888
106	3659	7034	7228	4985	4505	4107	5597	5619	3659	5004	6378	6202	6902	8080	7162	7705	8100	9499	15516	11400	29174	38805
107	4195	7773	7646	5232	4628	4195	5806	5628	4574	5368	6848	7056	7080	8676	7414	7951	8604	9740	16142	12486	32980	39968
108	4731	7827	8312	5409	4731	4879	6466	5924	5013	5721	7021	7080	7324	8892	8120	8209	8789	10625	16670	13161	34044	43524
109	4899	8018	8592	5840	4899	5111	6777	5927	5032	6343	7079	7384	7655	9389	8180	8382	9273	11537	16921	14569	38589	44212
110	5246	8677	8691	5953	5628	5289	6845	6446	5246	6496	7082	8421	8169	9911	8270	8984	9579	13578	17482	14871	38807	48563
111	5411	9729	8968	5996	5776	5781	7297	6544	5411	7681	7104	8435	8206	10875	8740	9833	9762	14621	18411	15173	38852	50387
112	5459	10591	9965	6159	5880	6233	7414	6576	5459	7731	7514	9443	8747	11026	8970	9956	15478	15501	19248	16264	40067	53183
113	5528	10677	10035	6213	5949	6453	7659	6578	5528	8302	8279	9927	9104	11183	9547	13610	16181	16539	20238	18277	40343	55558
114	5668	10683	10041	6218	6042	6468	8322	7091	5668	8419	8372	11102	9423	12804	13914	14968	16282	16717	22978	18476	40546	57334
115	6072	11556	11028	6872	6072	6733	8326	7510	6132	8946	8568	11194	10124	13772	15543	15027	17552	17404	23571	19399	40974	60400
116	6131	15029	11387	7388	6131	6852	9031	8150	6609	9478	8677	12485	10632	13886	15928	15335	17769	18416	23597	21391	41284	64575
117	6824	16189	12053	7516	6824	6961	9611	8458	7105	9897	8917	13728	10936	14388	16812	15489	17886	19506	23913	22958	41969	65949
118	7535	24392	13166	7557	8928	7535	10003	9084	8086	10065	10574	14333	10951	14957	17010	15820	18744	20541	24163	24591	42812	67007
119	7618	26228	13242	7799	10679	7618	10153	9625	8444	10215	12163	14900	12947	16304	17580	16002	21044	21058	24904	24987	50939	68339
120	7680	29394	13359	8412	11741	7680	10528	9777	9540	11358	12360	15350	18043	19028	18371	16107	22907	21424	26282	27198	55233	77362
121	8169	39390	14536	9439	12061	8169	11112	9859	10940	11367	12362	15480	19929	19860	20710	16173		23078	30904	27210	55539	79234
122	8642		14625	10161	13594	8642	11650	10533	11359	12389	13210	15845	20279	22758	21597	17801		24472	33685	31589	58648	80278
123	9502		18104	11609	14154	9502	12357	11369	11360	13208	13652	17787	20771	23256	21933	17883		25559	35756	32008	59824	82629
124	9506		18282	13511	17304	9506	12837	13175	11716	13464	14583	18456	22269	23457	22556	18184		27015	36148	32894	62689	87285
125	9981		18509	13517	20192	9981	12916	13983	12639	15452	16088	18548	23016	24082	23618	18641		27504	38794	34533	64679	90402
126	10332		22830	13674	20483	10332	13025	14385	13274	15745	17688	20412	23901	25003	28160	20824		27755	41242	36982	65516	96978
127	10848		25111	14560	20775	10848	13162	14453	14251	17819	19114	21549	24189	27467	31383	22279		28424	41792	37789	68311
128	11198		27308	16051	24046	11198	14574	14462	14820	18113	20503	22879	25223	31517	32904	22969		34351	44196	38521
129	12577		28986	16472	24120	12577	15860	14842	14886	18700	22211	23528	29096	33759	33100	23555		36933	44544	41216
130	12672		30957	17532	24396	12672	18142	15008	14902	19066	22887	27752	30467	35268	37209	25025		37822	45020	42162
131	13073		32318	17587	26136	13073	20931	15303	15589	19096	25002	28208	35060	36504	38416	25042		38396	49873	43760
132	13080		33221	19262	26830	13080	26468	17653	15982	21543	26252	31110	36510	39798	41502	25696		39864		48083
133	15043		34078	20000	28753	15043	27535	18144	20811	22956	26523	31253	38082	40745	48724	27460		41950		50358
134	16900		44589	20662	28596	16900	29052	25142	20863	23497	27251	32497	43141	45401		28895		45468		52282
135	19531		48283	21701	28925	19531	31521	25761	21014	26273	28233	34641	44753	46293		29104		51450		57222
136	20218			22884	29636	20218	33242	29403	22161	27050	29803	35248	48641	46454		29143		57938		66413
137	21960			27779	37499	21960	33836	34122	22396	27750	30173	36506	49677	49851		29808		58446		68512
138	22541			32018	42575	22541	35402	35326	22670	29100	31212	40242	56391	53357		30293		63706		72754
139	23034			32220	47388	23034	35755	39155	24430	29788	31424	43575	59791	56237		33118		63953		75541
140	24154			32967	48448	24154	37270	40441	24679	30037	32583	45364	62726	56248		34523		65113
141	24871			33865		27402	38683	44138	24871	32456	33696	54698	78211	57066		38647		67517
142	25771			38148		30612	41094	44763	25771	34887	36106	63384	78815	57442		39807
143	29135			41037		32912	41490	50213	29135	36679	36320	71207		66230		41365
144	29319			42456		33475	41563	50893	29319	37472	40172			66821		44472
145	35119			43490		36373	42239	53950	35119	38388	42457			71819		45690
146	35510			45494		36973	47069	67234	35510	40418	44900			73893		49501
147	36069			47136		38026	48702	72873	36069	41490	46170			75107		51039
148	37768					40007	52730	76577	37768	50698	46787			77583		56904
149	39663					42601	53444	77861	39663		47920			82052		58549
150	42842					42842	57873	78115	42846		49784			90954		60146
151	44096					50904	63931	80326	44096		53246					61970
152	44497					51658	71270	82317	44497		55579					63970
153	46270					53909	77127	92273	46270		59638					64199
154	47433					60193	79950	96914	47433		60260					68059
155	48365					61232	79974	105955	48365		64574					69898
156	51853					65240	80445	108330	51853		67190					75331
157	52967					66052	97018	117837	52967		67470					76853
158	53035					95621	103177	124578	53035		71221					86023
159	58573					97411	103445	139973	58573		73142					102954
160	59598					100438	106295	143571	59598		77776					112308
161	67574					103309		158755	67574		80678					112904
162	68587					110902		159331	68587		84684					125556
163	69612					114094		160412	69612		87557					131807
164	74332					128237			74332		96045					143989
165	77853					136696			77853		102231					146713
166	78530					145204			78530		102651					150515
167	81716					147339			81716		104202					156736
168	87299					165044			87299		104655					165557
169	95077					172883			95077		111235					180017
170	99889								99889		111239					191950
171	118689								118983		118689					192926
172	119254								136499		119254					211703
173	129630								139691		129630					216995
174	134337								140555		134337					242980
175	134490								141834		134490
176	141805								145565		141805
177	146149								153125		146149
178	159874								160854		159874
179	163330								181539		163330
180	168072								190248		168072
181	174712								205078		174712
182	177119								212228		177119
183	177684								218973		177684
184	190958								232755		190958
185	193804								239683		193804
186	197942								245214		197942
187	210616								260978		210616
188	226559								264131		226559
189	227776								273212		227776
190	229069										229069
191	245288										245288
192	255530										255530
193	294807										294807
194	318934										318934
195	334623										334623
196	334645										334645
197	363020										363020
198	376732										376732
199	403709										403709
200	414907										414907
201	449150										449150
202	472040										472040
203	479697										479697
204	496187										496187
205	498496										498496
206	517692										517692
207	531133										531133
208	549598										549598
209	587833										587833
210	608207										608207
211	608462										608462
212	639888										639888
213	716611										716611
214	788439										788439
215	834442										834442

Last fiddled with by robert44444uk on 2021-03-04 at 14:33

 2021-03-12, 12:35 #8 bur   Aug 2020 25·3 Posts That sounds interesting, strange that it was just discontinued. So apparently it's two parts, first computation of payam k and then doing the sieving/LLR with that k? I started payampentium using the supplied in.txt and it's doing something, but I have no clue what the values in in.txt mean. Will I just rediscover the k you already found 15 years ago? It created a recordtable.txt, which seems to contain the K(?) that is multiplied with 3*5*11*13*19*29*37*53 to yield the k and also says how many primes were found until which n. So now I have a list of good potential k-values, nice. Last fiddled with by bur on 2021-03-12 at 12:44
2021-03-13, 08:30   #9
robert44444uk

Jun 2003
Oxford, UK

78C16 Posts

Quote:
 Originally Posted by bur That sounds interesting, strange that it was just discontinued. So apparently it's two parts, first computation of payam k and then doing the sieving/LLR with that k? I started payampentium using the supplied in.txt and it's doing something, but I have no clue what the values in in.txt mean. Will I just rediscover the k you already found 15 years ago? It created a recordtable.txt, which seems to contain the K(?) that is multiplied with 3*5*11*13*19*29*37*53 to yield the k and also says how many primes were found until which n. So now I have a list of good potential k-values, nice.
The search was discontinued because many of the goals it set out to achieve were reached.

Yes it will rediscover K that provide very prime series. To get new results you need to scan a different area.

You will need to also set up a progress file where many variables are kept...progress.txt . The format can be downloaded at the software site.

c - 1 for Sierpinski series and -1 for Riesel series. i.e. K*M*2^n+1 and K*M*2^n-1

The E value is one less than the product of primes with primitive root 2 that you want to search - so 52 means 3*5*11*13*19*29*37*53

iteration Start this at 100000 to be sure you are looking at a fresh range. If you want to look at rich VPS areas don't look smaller than E = 82. You would get 10-20 or so VPS results in a day.

I this is a subcategory of iteration to ensure the program can pick up where it left off

I'm using payam2.exe as it allows for smaller E to be used and is faster.

In the in.txt file you will see any number of variables, but you don't need to change them for smaller M. You can switch off the smith_check indicator by using 0 as the flag for larger M, which will provide a more detailed search but perhaps not pick up the better values as quickly. For very large M, say 268 onwards, then you also need to reduce vps count to say 10, as candidates only pop up twice a day for 268 and once in a blue moon for higher numbers. I have found 1 payam number at 316 and 1 at 292 after 5 days of searching, but the 316 is a fluke - you could go months without finding a payam number. No-one ever found a 346 in the years of searching.

I've decided to look at the Sierpinski side starting everything from 100000. My best result after 2 days of searching is 114 primes in 10000 n.

The reason I wanted to make this multithread is to really speed up the search.

 2021-03-14, 17:15 #10 bur   Aug 2020 25·3 Posts Ok, so the goal was finding k that produced a large number of primes for small n and not so much identifying one or two such k and then doing a thorough search for n up to a few million? I wonder if the prime density remains that high even when n ~ 1e6. And more importantly, if even after sieving the prime density would remain higher than for other k.
2021-03-15, 18:13   #11
robert44444uk

Jun 2003
Oxford, UK

22·3·7·23 Posts

Quote:
 Originally Posted by bur Ok, so the goal was finding k that produced a large number of primes for small n and not so much identifying one or two such k and then doing a thorough search for n up to a few million? I wonder if the prime density remains that high even when n ~ 1e6. And more importantly, if even after sieving the prime density would remain higher than for other k.
No one k was taken up as far as n=1e6, about 850k from memory was the highest. That k has more than 200 primes, and is the only known k with >200 primes.

Checking is quite slow for these k, because there is no really superfast sieve (although one could be constructed!), newpgen is not really geared from today's computers.

Also, because there are no small factors for any of the k*2^n+ or -1, a significantly greater number of n need a prp test. Hence a payam k series will remain, on average, more prime at any level of n compared to a random k because of this.

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