Depression and Plateau Primes

4*(10^11-1)/9 + 3*(10^10+1)

run"aprt-cle
Test number  N=? 4*(10^11-1)\9 + 3*(10^10+1)

Preparatory test
    Pass !

Main test  for P= 2
   for Q= 3    for Q= 5    for Q= 7    for Q= 13

Main test  for P= 3
   for Q= 7    for Q= 13

Main test  for P= 5
    Pass !

 74444444447 is prime.
  0:00:00
OK

Proved prime with 'Ubasic - APRT-CLE.UB'
by Patrick De Geest using a Pentium III 650 MHz chip.

*4(10^31-1)/9 + 3(10^30+1)*
== ID:B28650104AC4E ============================================= PRIMO 1.2.1 - Primality Certificate ----------------------------------------------------------------- Candidate ----------------------------------------------------------------- N = 7444444444444444444444444444447 Decimal size = 31 Binary size = 103 ----------------------------------------------------------------- 1) EC Test ----------------------------------------------------------------- N = Candidate S = 43579 R = 170826417413076014195584393 A = 0 B =-1861111111111111111111111111112 T = 3 ----------------------------------------------------------------- 2) N+1 Test ----------------------------------------------------------------- N = R of preceding test S = 806 R = 211943445921930538704199 Q = 11 ----------------------------------------------------------------- 3) N-1 Test ----------------------------------------------------------------- N = R of preceding test S = 219126 R = 967221808100958073 B = 2 ----------------------------------------------------------------- 4) N-1 Test ----------------------------------------------------------------- N = R of preceding test S = 4093752 R = 236267807161 B = 2 ----------------------------------------------------------------- 5) SPP Test ----------------------------------------------------------------- N = R of preceding test Started 01/22/2003 04:44:43 AM Running time < 1s Candidate certified prime ================================================================= Proved prime with 'Primo 1.2.1' by Patrick De Geest using a Pentium III 650 MHz chip.

4*(10^31-1)/9 + 3*(10^30+1)


== ID:B28650104AC4E =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 7444444444444444444444444444447

Decimal size = 31
Binary size = 103

-----------------------------------------------------------------
1) EC Test
-----------------------------------------------------------------
N = Candidate
S = 43579
R = 170826417413076014195584393
A = 0
B =-1861111111111111111111111111112
T = 3

-----------------------------------------------------------------
2) N+1 Test
-----------------------------------------------------------------
N = R of preceding test
S = 806
R = 211943445921930538704199
Q = 11

-----------------------------------------------------------------
3) N-1 Test
-----------------------------------------------------------------
N = R of preceding test
S = 219126
R = 967221808100958073
B = 2

-----------------------------------------------------------------
4) N-1 Test
-----------------------------------------------------------------
N = R of preceding test
S = 4093752
R = 236267807161
B = 2

-----------------------------------------------------------------
5) SPP Test
-----------------------------------------------------------------
N = R of preceding test

Started 01/22/2003 04:44:43 AM
Running time < 1s

Candidate certified prime

=================================================================

Proved prime with 'Primo 1.2.1'
by Patrick De Geest using a Pentium III 650 MHz chip.

4*(10^121-1)/9 + 3*(10^120+1)

== ID:B2869036D92C0 =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 744444444444444444444444444444444444444444444444444444444444\
    444444444444444444444444444444444444444444444444444444444444\
    7

Decimal size = 121
Binary size = 402


Started 01/26/2003 03:58:32 PM
Running time 1s

Candidate certified prime

=================================================================

Proved prime with 'Primo 1.2.1'
by Patrick De Geest using a Pentium III 650 MHz chip.

4*(10^485-1)/9 + 3*(10^484+1)

== ID:B286C011BFD5E =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 744444444444444444444444444444444444444444444444444444444444\
    444444444444444444444444444444444444444444444444444444444444\
    444444444444444444444444444444444444444444444444444444444444\
    444444444444444444444444444444444444444444444444444444444444\
    444444444444444444444444444444444444444444444444444444444444\
    444444444444444444444444444444444444444444444444444444444444\
    444444444444444444444444444444444444444444444444444444444444\
    444444444444444444444444444444444444444444444444444444444444\
    44447

Decimal size = 485
Binary size = 1611


Started 01/29/2003 05:10:11 AM
Running time 10mn 57s

Candidate certified prime

=================================================================

Proved prime with 'Primo 1.2.1'
by Patrick De Geest using a Pentium III 650 MHz chip.

4*(10^1487-1)/9 + 3*(10^1486+1)

[PRIMO - Primality Certificate]
Version=2.1.1
WebSite=http://www.ellipsa.net/
Format=3
ID=B290800AFB284
Created=07/04/2003 03:11:59 AM
TestCount=217
Status=Candidate certified prime

[Running Times]
Initialization=5.31s
1stPhase=6h 24mn 14s
2ndPhase=39mn 58s
Total=7h 4mn 17s

[Candidate]
File=C:\Primo IN & OUT files\PDP1487.in
Expression=4*(10^1487-1)/9+3*(10^1486+1)
N$=9B8EF8411F5817EC83D518A6E5165AE50752691C467967A70FEBD89746EE4E6C5196A93D85F0D61E08FBE3D75AC3C79F\
0DF7950E8A8E0A8140B23B37C6DD044ECC38BBB11424C58A0C0D0E59470AB14E670EF251AA6A5DB708C95083C772E092831\
C1E469B79D312A3296413B607F82BD47F1BCAB54B52B449E9BCC7C3495646EBF81170D16125C6778D78CE24647EB1225005\
F8A957A88636244750F6243B364EDE03A02064C9814C433D9FFF2BC7CBA8F09493389B88AD1B369F387D302D5304E9BED55\
530BABAF2259BE1EF8104B883C72CB3E00C68CCF34258D17A3F013ED8D376DBA1C97C953268027C1AC5C3B1307B9A7725D4\
FB1149E8832AC1FD06CE27F48DD4A882A4A1BCFD62B6B46770E7A8BB1B99683E1F4C99D16F81A2E5E3A551330B693968BAE\
26127C86F65BEB504D811E8C0FB4119D02A88D4A24C14F61D7CFEFE124A61C8312C0ACC4FE258B936BD94FCB58FF360F916\
4F776015208B95C362FBCC7E0D24F1C6B9B51A24BA704462FD8AD926A4D352BAF890542E2D1103AC75E9287297ECE2D2E34\
8DF6F7FA6249E5E58BC1FFEC840EF1052FCD6E9218E29A9A732029AB50FE5E2D3AA3C2741531C71C71C71C71C71C71C71C7\
1C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C7\
1C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C7\
1C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C7\
1C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71F
HexadecimalSize=1235
DecimalSize=1487
BinarySize=4940

Proved prime with 'Primo 2.1.1'
by Patrick De Geest using a 3000 MHz Pentium 4 cpu.
Certificate Primo-B290800AFB284-01.out available by
simple email request (249 KB).

4*(10^1579-1)/9 + 3*(10^1578+1)

[PRIMO - Primality Certificate]
Version=2.1.1
WebSite=http://www.ellipsa.net/
Format=3
ID=B290A02E8F8A5
Created=07/06/2003 01:33:46 PM
TestCount=236
Status=Candidate certified prime

[Running Times]
Initialization=3.63s
1stPhase=9h 15mn 7s
2ndPhase=57mn 0s
Total=10h 12mn 11s

[Candidate]
File=C:\Primo IN & OUT files\PDP1579.in
Expression=4*(10^1579-1)/9+3*(10^1578+1)
N$=1DD482CE2D0D9153E19CC510278D1E818B40FBE0CAB334B6578B8294A8FFCA69B1DB2E7D20DC5790D13C45498A7EB971\
10A51F115CEE6BA5FDDC93456C849616268EECB433EF7C4F77383437F5EE611D84BF7358834C0DFD0C7F2BAB986B8494743\
D2CCE6178D17EA1D20A2268F1D27814D9C4983FBCFEEF47225F883415FFFDB9C1AD92D21C781CBF1A23A8EF2B664F3B3F8D\
B12629F11489B604642365BBD97C15288642BAF3A472538E2164389CCB30F2A4B09D3499909783AC0C7BACA41BF8BF5AC99\
89F7F0547405830D0066F16A254504503E93348E331E70C7ED6425378B5F6859B55BB782C7BDC5706E3E66D1BBC3AC13BE3\
44938DE99D313363D38FB1025DD60414EC64CA0E04138F7E9A537FAAD9436397975615DEED3FF83D665123448FBFB8D7FC5\
728B7608B4D587801AE3B69224262B9948CC9A73D16A17FBFBEF08B1D1B3DBF8BFBAC3DC66B9527A4969235B6848AAFAC69\
F948EF45481DC36EB2B039621E18C175A2FA077984AB8CF1188DCFCD7663D7875DC7FD9D36EA89E6EA2A965547ADEB8977B\
CB1FCCB38CE34CBEA716F9149DE7F9564BF0FC0426DD3A67A3F6EE703756C76DAF48C98E2C9F20138E769562108C1E3F98C\
7AF8EDE8463CC0338ED83F44FEE0BDC71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71\
C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71\
C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71\
C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71C71\
C71C71C71C71C71C71C71C71C71F
HexadecimalSize=1312
DecimalSize=1579
BinarySize=5245

Proved prime with 'Primo 2.1.1'
by Patrick De Geest using a 3000 MHz Pentium 4 cpu.
Certificate Primo-B290A02E8F8A5-01.out available by
simple email request (287 KB).

4*(10^13673-1)/9 + 3*(10^13672+1)

3-PRP!

4*(10^13811-1)/9 + 3*(10^13810+1)

3-PRP

4*(10^15095-1)/9 + 3*(10^15094+1)

3-PRP!

4*(10^72773-1)/9 + 3*(10^72772+1)

PRP tested by Ray Chandler
pfgw -q"(67*10^72772+23)/9" -tc

PFGW Version 3.4.2.64BIT.20101019.Win_Dev [GWNUM 26.4] 
 
Primality testing (67*10^72772+23)/9 [N-1/N+1, Brillhart-Lehmer-Selfridge] 
Running N-1 test using base 5 
Running N+1 test using discriminant 19, base 1+sqrt(19) 
Calling N-1 BLS with factored part 0.01% and helper 0.00% (0.02% proof) 
(67*10^72772+23)/9 is Fermat and Lucas PRP! (1645.1859s+0.0020s)

4*(10^94213-1)/9 + 3*(10^94212+1)

PRP tested by Ray Chandler

PFGW Version 3.4.5.64BIT.20110215.Win_Dev [GWNUM 26.5] 
 
Primality testing (67*10^94212+23)/9 [N-1/N+1, Brillhart-Lehmer-Selfridge] 
Running N-1 test using base 3 
Running N+1 test using discriminant 67, base 2+sqrt(67) 
Calling N-1 BLS with factored part 0.01% and helper 0.01% (0.05% proof) 
(67*10^94212+23)/9 is Fermat and Lucas PRP! (3045.3945s+0.0032s)

4*(10^207557-1)/9 + 3*(10^207556+1)

By Serge Batalov dd. January 13, 2023.

PRP Source : https://stdkmd.net/nrr/7/74447.htm#prime

4*(10^1116677-1)/9 + 3*(10^1116676+1)

By Ryan Propper & Serge Batalov dd. January 22, 2023.

PRP announced at https://www.mersenneforum.org/showthread.php?t=28425

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Patrick De Geest - Belgium

- Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com