Henri & Renaud Lifchitz's PRP Top records

What is a PRP ?

A PRP is a probable prime number, a number that nobody knows how to prove or disprove its primality.
Here are some criterions that can be used to recognize these numbers :

- N does not have any prime factor up to 2^32.
- N cannot be trivially written into a product.
- Take 3 square-free consecutive bases such that (b1/N)=(b2/N)=(b3/N)= -1. N should be such that b1^((N-1)/2)=b2^((N-1)/2)=b3^((N-1)/2)= -1 (mod N).
- Take a prime base such that (b/N)= -1 and b<>b1<>b2<>b3. N should pass the strong test in base b.
- or more (if someone has got an idea) !!!

Submit your PRPs

See the Top discoverers
See the Latest PRPs
Search by form


1-250	251-500	501-750	751-1000	1001-1250	1251-1500	1501-1750	1751-2000
2001-2250	2251-2500	2501-2750	2751-3000	3001-3250	3251-3500	3501-3750	3751-4000
4001-4250	4251-4500	4501-4750	4751-5000	5001-5250	5251-5500	5501-5750	5751-6000
6001-6250	6251-6500	6501-6750	6751-7000	7001-7250	7251-7500	7501-7750	7751-8000
8001-8250	8251-8500	8501-8750	8751-9000	9001-9250	9251-9500	9501-9750	9751-10000

Rank	Probable prime	Digits (> 50000)	Who ?	When ?
9001	(26*10^104132-503)/9	104133	Robert Price	07/2018
9002	3*5^148968+8	104125	Lelio R Paula	11/2014
9003	(7*10^104113+53)/3	104114	Robert Price	05/2018
9004	p(21253)#/2+8	104113	Robert Price	01/2017
9005	33582^1259+1259^33582	104106	Hans Havermann	09/2020
9006	32958^1441+1441^32958	104104	Hans Havermann	09/2020
9007	29127^3748+3748^29127	104095	Hans Havermann	09/2020
9008	322^41507+41507^322	104094	Norbert Schneider	12/2014
9009	6^133768-17423	104092	Enzo Doria	03/2013
9010	2^345786-19557	104092	Enzo Doria	05/2013
9011	171^46606-2	104072	Lelio R Paula	10/2017
9012	(29*10^104071+91)/3	104072	Robert Price	04/2020
9013	norm(Mod(x,x^3-x-1)^852167-1)	104070	Paul Vanderveen	09/2022
9014	2^345704-149	104068	Lelio R Paula	09/2011
9015	25821^10712+10712^25821	104056	Hans Havermann	09/2020
9016	(2^345662+2038)/2	104055	Stefano Morozzi	09/2020
9017	6^133704-17273	104042	Enzo Doria	02/2013
9018	(3*2^345590-1)/23	104033	Lelio R Paula	03/2014
9019	(19*2^345590+1)/33859	104030	Lelio R Paula	12/2013
9020	2^345547+11	104021	Lelio R Paula	09/2011
9021	(19*10^104014-49)/3	104015	Robert Price	05/2019
9022	(2^345519+1794)/2	104012	Stefano Morozzi	01/2021
9023	2^345504+723	104008	Lelio R Paula	09/2011
9024	2^345498-20037	104006	Enzo Doria	06/2013
9026	18^82854-37	104005	Lelio R Paula	01/2017
9025	(2^345497+2070)/2	104005	Stefano Morozzi	09/2020
9027	2^345486-109065	104002	Enzo Doria	04/2014
9028	2^345487-1851507	104002	Enzo Doria	08/2015
9029	2^345476+777	103999	Lelio R Paula	09/2011
9030	(2^345477+1554)/2	103999	Stefano Morozzi	09/2020
9031	4*10^103981-83	103982	Robert Price	10/2018
9032	(2^345327+3206)/2	103954	Stefano Morozzi	09/2020
9033	(5*2^345321-1)/3	103953	Lelio R Paula	03/2014
9034	(2^345267-9)/539049127	103927	Lelio R Paula	12/2013
9035	(19*10^103907+77)/3	103908	Robert Price	05/2019
9036	(78557*2^345163+1)/78589	103905	Lelio R Paula	07/2014
9037	2^345106-112181	103888	Enzo Doria	06/2014
9038	(2^345105+1326)/2	103887	Stefano Morozzi	02/2021
9039	2^345079+99	103880	Lelio R Paula	09/2011
9040	(2^345034+1198)/2	103866	Stefano Morozzi	09/2020
9041	11^99736-1500	103865	Enzo Doria	12/2012
9042	(569^37699-1)/568	103863	Serge Batalov	06/2022
9043	(2^345010+2082)/2	103859	Stefano Morozzi	09/2020
9044	23263^23784-23784^23263	103857	Norbert Schneider	01/2020
9045	3*10^103844+31	103845	Robert Price	06/2021
9046	444*33^68368+19	103821	Eric Chen	05/2023
9047	23927^21814+21814^23927	103813	Hans Havermann	09/2020
9048	2^344842-20157	103808	Enzo Doria	06/2013
9049	(47*2^344822+1)/1071	103801	Lelio R Paula	05/2015
9050	2^344802-20765	103796	Enzo Doria	06/2013
9051	2^344794-3821	103794	Yves Minet	05/2011
9052	2^344782-109103	103790	Enzo Doria	04/2014
9053	3^217521-112292	103784	Enzo Doria	06/2014
9054	2^344759-3505	103783	Lelio R Paula	11/2016
9055	2^344735-147	103776	Lelio R Paula	09/2011
9056	L(496533)-p(496533)	103770	Robert Price	02/2018
9057	2*10^103749-69	103750	Robert Price	03/2018
9058	(55459*2^344617+1)/3	103745	Lelio R Paula	07/2014
9059	2^344607-108891	103738	Enzo Doria	03/2014
9061	(2^344608+3798)/2	103738	Stefano Morozzi	02/2021
9060	(1097^34123-1)/1096	103738	Serge Batalov	06/2022
9062	2^344560+22411	103723	Enzo Doria	09/2023
9063	(148*10^103718-1)/3	103720	Robert Price	03/2020
9064	2^344541-19561	103718	Enzo Doria	05/2013
9065	(1017*2^344529-1)/569	103714	Lelio R Paula	05/2015
9066	(8896^26263-1)/8895	103714	Serge Batalov	06/2022
9067	(2^344503+1758)/2	103706	Stefano Morozzi	09/2020
9068	(4*10^103703-1)/3	103704	Serge Batalov	08/2009
9069	2^344444-455	103688	Yves Minet	03/2012
9070	(4*10^103678-43)/3	103679	Robert Price	01/2022
9071	(1353^238539617^9)/10^307661	103673	Hans Havermann	06/2017
9072	(2^344367+2738)/2	103665	Stefano Morozzi	01/2021
9074	2^344317-109933	103650	Enzo Doria	04/2014
9073	26055^9508+9508^26055	103650	Hans Havermann	09/2020
9075	2^344309-1852033	103648	Enzo Doria	08/2015
9076	25729^10672+10672^25729	103643	Hans Havermann	09/2020
9077	(2^344289+1166)/2	103642	Stefano Morozzi	09/2020
9078	3*10^103637+61	103638	Robert Price	09/2020
9079	(2^344246+1614)/2	103629	Stefano Morozzi	09/2020
9080	2^344213-21253	103619	Enzo Doria	06/2013
9081	2^344196-465	103614	Lelio R Paula	11/2016
9082	2^344187-20001	103611	Enzo Doria	06/2013
9083	(117191#+(355573#)/(117191#))/(200233679)	103602	Makoto Morimoto	10/2017
9084	(237^43627-1)/236	103601	Serge Batalov	06/2022
9085	20^79621-19^79621	103590	Jean-Louis Charton	03/2011
9086	Phi(155391,10)/1243129	103586	Maksym Voznyy	09/2015
9087	(9^108553-2^108553)/7	103586	Jon Grantham	07/2023
9088	2^344085-20365	103580	Enzo Doria	06/2013
9089	5^148169+54	103566	Lelio R Paula	11/2016
9090	2^344017-108795	103560	Enzo Doria	03/2014
9091	(2^343974+2670)/2	103547	Stefano Morozzi	09/2020
9092	2^343971-5077	103546	Lelio R Paula	09/2011
9093	(7*10^103531+71)/3	103532	Robert Price	05/2018
9094	2^343899+141	103524	Lelio R Paula	08/2011
9095	(2^343933+1)/(3*98294339989393)	103520	Gord Palameta	08/2018
9096	6^133029-67	103517	Lelio R Paula	11/2016
9097	(17*2^343877+1)/13625	103515	Lelio R Paula	12/2013
9098	66*10^103507+7	103509	Robert Price	07/2020
9099	26297^8614+8614^26297	103485	Hans Havermann	09/2020
9100	(2^343764+2658)/2	103483	Stefano Morozzi	09/2020
9101	26499^8036+8036^26499	103480	Hans Havermann	09/2020
9102	(103^51413-1)/4478483706	103477	Paul Vanderveen	06/2025
9103	(136*10^103474-1)/9	103476	Serge Batalov	12/2014
9104	5^148014-24	103458	Lelio R Paula	11/2016
9105	25^74007-24	103458	Anonymous	09/2024
9106	28316^4503+4503^28316	103453	Hans Havermann	09/2020
9107	(22*10^103450+53)/3	103451	Robert Price	07/2019
9108	2^343604+9931	103436	Yves Minet	01/2011
9109	2^343601-21189	103435	Enzo Doria	06/2013
9110	(10^103423+173)/3	103423	Robert Price	05/2021
9111	2^343508-115229	103407	Enzo Doria	06/2015
9112	(8766^26227-1)/8765	103404	Serge Batalov	06/2022
9113	(509203*2^343491-1)/16147	103403	Lelio R Paula	05/2015
9114	3^216685-113056	103386	Enzo Doria	07/2014
9115	24829^14580+14580^24829	103382	Hans Havermann	08/2020
9116	91355!4+1024	103381	Robert Price	09/2019
9117	44^62903-43^62903	103379	Jean-Louis Charton	08/2012
9118	11^99262-20	103371	Lelio R Paula	12/2016
9119	(509203*2^343357-1)/5	103366	Lelio R Paula	05/2015
9120	(2^343363+3570)/2	103363	Stefano Morozzi	09/2020
9121	2^343356-2791889	103361	Enzo Doria	03/2016
9122	(25*10^103360-241)/9	103361	Robert Price	05/2018
9123	Phi(104989,10)/2939693	103350	Maksym Voznyy	09/2015
9124	(371*33^68040-19)/32	103321	Eric Chen	05/2023
9125	24826^14511+14511^24826	103319	Hans Havermann	09/2020
9126	2^343182-171	103309	Lelio R Paula	08/2011
9127	(23619^23619+23620^23620)/7187	103294	David Broadhurst	09/2009
9128	(16487^24649-1)/(16487^157-1)	103287	Ryan Propper	11/2023
9129	(10680^25639-1)/10679	103285	Serge Batalov	06/2022
9130	(7427^26683-1)/7426	103282	Serge Batalov	06/2022
9131	2^343079-1835611	103278	Enzo Doria	05/2015
9132	2^343063-1147	103273	Yves Minet	06/2013
9133	24741^14930+14930^24741	103271	Hans Havermann	09/2020
9134	(16*10^103255-73)/3	103256	Robert Price	07/2020
9135	(219^44119-1)/218	103256	Serge Batalov	06/2022
9136	9*6^132689+7	103254	Lelio R Paula	11/2014
9137	2^342979-18085	103247	Enzo Doria	05/2013
9138	(16*10^103244-31)/3	103245	Robert Price	03/2019
9139	2^342966-16995	103244	Lelio R Paula	11/2016
9140	23797^21774+21774^23797	103230	Hans Havermann	09/2020
9141	25388^11637+11637^25388	103224	Hans Havermann	09/2020
9142	(2^342876+2086)/2	103216	Stefano Morozzi	09/2020
9143	(16*10^103214-31)/3	103215	Robert Price	03/2019
9145	(37*10^103211+611)/9	103212	Robert Price	11/2018
9144	(238*10^103210-1)/3	103212	Robert Price	04/2020
9146	5^147650+16	103203	Lelio R Paula	11/2016
9147	gcd(Phi(852060,10),(10^42603+1)^3(10^42603+1-10^21302)+(10^85206+10^42603)(10^21302+1)+10^127809)	103201	Serge Batalov	04/2015
9148	Phi(852060L,10)	103201	Anonymous	03/2025
9149	2^342817+2707067	103199	Enzo Doria	03/2016
9150	43^63177-42	103198	Theodore Burton	02/2007
9151	2^342805-21025	103195	Enzo Doria	06/2013
9152	24129^18890+18890^24129	103182	Hans Havermann	09/2020
9153	(622*10^103171-1)/9	103173	Robert Price	11/2021
9154	2^342714+12285	103168	Lelio R Paula	11/2016
9155	(2*10^103161-17)/3	103161	Serge Batalov	11/2022
9156	18^82178-4481	103156	Enzo Doria	03/2013
9157	2^342665-17349	103153	Enzo Doria	05/2013
9158	30408^2467+2467^30408	103150	Hans Havermann	09/2020
9159	(5^147571-3^147571)/2	103148	Kellen Shenton	05/2022
9160	(2^342615+3090)/2	103138	Stefano Morozzi	09/2020
9161	2^342612+175	103137	Lelio R Paula	08/2011
9162	5*2^342585-3	103130	Lelio R Paula	09/2014
9163	(16*10^103127+77)/3	103128	Robert Price	03/2019
9164	(2^342527+3774)/2	103111	Stefano Morozzi	02/2021
9165	2^342503-19209	103104	Enzo Doria	05/2013
9166	(2^342484+2586)/2	103098	Stefano Morozzi	09/2020
9167	2^342476+51	103096	Lelio R Paula	08/2011
9168	2^342460-18885	103091	Enzo Doria	05/2013
9169	(2^342442+862)/2	103086	Stefano Morozzi	02/2021
9170	(194*10^103074-17)/3	103076	Robert Price	09/2020
9171	7^121968+82	103075	Lelio R Paula	10/2016
9172	2^342392-108863	103071	Enzo Doria	03/2014
9173	(2^342389+806)/2	103070	Stefano Morozzi	01/2021
9174	3^216014+28	103065	Lelio R Paula	11/2016
9175	2^342353-108879	103059	Enzo Doria	03/2014
9176	10^103055+276081	103056	Patrick De Geest	12/2016
9177	10^103055-53	103055	Lelio R Paula	11/2016
9178	(58*10^103046+419)/9	103047	Robert Price	05/2019
9179	7^121933+4	103046	Lelio R Paula	11/2014
9180	(71*10^103028-143)/9	103029	Robert Price	10/2021
9181	(2^342235+3618)/2	103023	Stefano Morozzi	09/2020
9182	19^80560-68	103017	Lelio R Paula	10/2016
9183	2^342209+76607	103016	Enzo Doria	05/2016
9185	328^40945+40945^328	103013	Norbert Schneider	12/2014
9184	12^95454-91	103013	Lelio R Paula	10/2016
9186	25833^9716+9716^25833	103009	Hans Havermann	01/2019
9187	(893*10^103004+7)/9	103006	Robert Price	06/2021
9188	(1475^32503-1)/1474	102993	Serge Batalov	06/2022
9189	65^56809-64^56809	102990	Robert Price	08/2013
9190	5^147338-6	102985	Lelio R Paula	11/2014
9191	2^342084-112739	102978	Enzo Doria	06/2014
9192	9*6^132330+7	102974	Lelio R Paula	11/2014
9193	2^342034-16737	102963	Enzo Doria	05/2013
9194	(9787*2^342011-1)/25	102959	Lelio R Paula	05/2015
9195	(3^215802-5)/124343756	102956	Lelio R Paula	12/2013
9196	2^341991-17421	102950	Enzo Doria	05/2013
9197	2^341969-112225	102943	Enzo Doria	06/2014
9198	(9787*2^341864-1)/3	102915	Lelio R Paula	05/2015
9199	2^341859+1869413	102910	Enzo Doria	05/2016
9200	10^102909+61	102910	Lelio R Paula	11/2016
9201	10^102909-67049	102909	Patrick De Geest	12/2016
9202	23814^20935+20935^23814	102898	Hans Havermann	02/2019
9203	28345^4264+4264^28345	102888	Hans Havermann	02/2019
9204	25665^10204+10204^25665	102886	Hans Havermann	02/2019
9205	26288^8181+8181^26288	102860	Hans Havermann	02/2019
9206	2^341659+1836773	102850	Enzo Doria	03/2016
9207	2^341608-110639	102835	Enzo Doria	05/2014
9208	10^102824+49	102825	Lelio R Paula	11/2016
9209	10^102824-168011	102824	Patrick De Geest	12/2016
9210	(156^46877-1)/155	102805	Anonymous	06/2024
9211	11^98708-8	102794	Lelio R Paula	10/2016
9212	2^341466+4429	102792	Lelio R Paula	09/2011
9213	2^341460-107	102790	Lelio R Paula	08/2011
9214	(2^341388+47)/21014943	102761	Lelio R Paula	12/2013
9215	2^341355+21	102759	Lelio R Paula	08/2011
9216	2^341319+224885	102748	Enzo Doria	07/2024
9217	(419^39181-1)/418	102739	Tom Wu	07/2024
9218	(6496^26947-1)/6495	102736	Serge Batalov	06/2022
9219	2^341246+40755	102726	Lelio R Paula	11/2016
9220	(2^341256-5)/991699	102723	Lelio R Paula	12/2013
9221	7^121547+78	102720	Lelio R Paula	10/2016
9222	2^341223-461197	102719	Enzo Doria	06/2015
9223	27838^4893+4893^27838	102711	Hans Havermann	03/2019
9225	2^341190-20577	102709	Enzo Doria	06/2013
9224	(27*2^341212+1)/84674449	102709	Lelio R Paula	12/2013
9226	(25*10^102698+77)/3	102699	Robert Price	09/2022
9227	6^131976-65	102698	Lelio R Paula	10/2016
9228	2^341134+2533	102692	Henri Lifchitz	09/2005
9229	13^92184-54	102688	Lelio R Paula	10/2016
9230	2^341116-20873	102687	Enzo Doria	06/2013
9231	2^341094-19953	102680	Enzo Doria	06/2013
9232	23761^20950+20950^23761	102676	Hans Havermann	03/2019
9234	2^341053+129	102668	Lelio R Paula	08/2011
9233	(4724^27943-1)/4723	102668	Serge Batalov	06/2022
9236	(31*2^341006-1)/3	102655	Lelio R Paula	03/2014
9235	7^121471+94	102655	Lelio R Paula	10/2016
9238	2^341004-173	102653	Lelio R Paula	08/2011
9237	20^78901+3	102653	Lelio R Paula	10/2016
9239	(2^341001+1674)/2	102652	Stefano Morozzi	09/2020
9240	5^146853-52	102646	Lelio R Paula	10/2016
9241	2^340969-216931	102642	Enzo Doria	05/2014
9242	122^49196+121	102641	Norbert Schneider	10/2018
9243	2^340952+1870021	102637	Enzo Doria	05/2016
9244	5*2^340940-3	102634	Lelio R Paula	08/2014
9245	(6^131891+1)/7	102631	Paul Bourdelais	01/2010
9246	(11*10^102629-23)/3	102630	Robert Price	09/2021
9247	3^215094+5*2^215094	102626	Jon Grantham	08/2023
9248	25876^9239+9239^25876	102615	Hans Havermann	02/2019
9249	(4*10^102612+83)/3	102613	Robert Price	02/2018
9250	(234^43313-1)/20184091	102611	Aurel Gibier	03/2025

Find by discoverer :

PRP Records

Probable Primes Top 10000

(302273 PRPs in the top)