Depression and Plateau Primes

2*(10^3-1)/9 + 5*(10^2+1)

*** VFYPR 1.13F  F_max=100000 S_min=100000 h=0 a=0 C=0 J=0 D=0 
N=(72*10^3-27)/99
N=727
*** N is prime!
Time: 0 sec

2*(10^5-1)/9 + 5*(10^4+1)

run"aprt-cle
Test number  N=? 72227

Preparatory test
    Pass !

 72227 is prime.
  0:00:00
OK

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

2*(10^9-1)/9 + 5*(10^8+1)

run"aprt-cle
Test number  N=? 722222227

Preparatory test
    Pass !

 722222227 is prime.
  0:00:00
OK

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

*2(10^29-1)/9 + 5(10^28+1)*
== ID:B286500F8D284 ============================================= PRIMO 1.2.1 - Primality Certificate ----------------------------------------------------------------- Candidate ----------------------------------------------------------------- N = 72222222222222222222222222227 Decimal size = 29 Binary size = 96 ----------------------------------------------------------------- 1) N-1 Test ----------------------------------------------------------------- N = Candidate S = 24178 R = 2987104897932923410630417 B = 2 ----------------------------------------------------------------- 2) N+1 Test ----------------------------------------------------------------- N = R of preceding test S = 2 R = 1493552448966461705315209 Q = 14 ----------------------------------------------------------------- 3) EC Test ----------------------------------------------------------------- N = R of preceding test S = 7101297 R = 210321079229440261 A = 0 B = 16 T = 2 ----------------------------------------------------------------- 4) N-1 Test ----------------------------------------------------------------- N = R of preceding test S = 73980 R = 2842945109887 B = 2 ----------------------------------------------------------------- 5) SPP Test ----------------------------------------------------------------- N = R of preceding test Started 01/22/2003 04:31:46 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.

2*(10^29-1)/9 + 5*(10^28+1)


== ID:B286500F8D284 =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 72222222222222222222222222227

Decimal size = 29
Binary size = 96

-----------------------------------------------------------------
1) N-1 Test
-----------------------------------------------------------------
N = Candidate
S = 24178
R = 2987104897932923410630417
B = 2

-----------------------------------------------------------------
2) N+1 Test
-----------------------------------------------------------------
N = R of preceding test
S = 2
R = 1493552448966461705315209
Q = 14

-----------------------------------------------------------------
3) EC Test
-----------------------------------------------------------------
N = R of preceding test
S = 7101297
R = 210321079229440261
A = 0
B = 16
T = 2

-----------------------------------------------------------------
4) N-1 Test
-----------------------------------------------------------------
N = R of preceding test
S = 73980
R = 2842945109887
B = 2

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

Started 01/22/2003 04:31:46 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.

2*(10^65-1)/9 + 5*(10^64+1)

== ID:B2869006422BA =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 722222222222222222222222222222222222222222222222222222222222\
    22227

Decimal size = 65
Binary size = 216


Started 01/26/2003 01:49:22 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.

2*(10^725-1)/9 + 5*(10^724+1)

[PRIMO - Primality Certificate]
Version=2.0.0 - beta 3
Format=2
ID=B286D03FF84FA
Created=01/30/2003 06:37:57 PM
TestCount=119
Status=Candidate certified prime

[Candidate]
File=C:\Program Files\Primo200\pdp727_725.in
Expression=2*(10^725-1)/9 + 5*(10^(723+1)+1)
N$=F39A2B868857B15B4E2BA984631ABC0A898799AEC2595F45FF37EA197F3DDA3E30A3FD6B5A6BFBA95E4D33782763FC43482726C152E8307B4FA50E2BEEF046103AC1FFD5E98365B3A6D124ECDE7B1C0882C173586147283D5854C5280420EDE36F487A0BB141ED14D10513412EE752FEE493C1D79CF1B19203BE5E6FE8352F0FC4240977AB93366C4518EFD2857914A21FF382E64276F717C2350EA8CA1589BC7A9EC4A5D11E3410E82A06C29F0D99A81077211CC3CE4DCE8CBA17B67597AE861F015DF3BDE3AB4E7CACCFB96317A916FC461E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E393
HexadecimalSize=602
DecimalSize=725
BinarySize=2408

[Running Times]
Initialization=1.44s
1stPhase=1h 35mn 36s
2ndPhase=17mn 20s
Total=1h 52mn 58s

Proved prime with 'Primo 2.0.0 - beta 3'
by Patrick De Geest using a Pentium III 650 MHz chip.

2*(10^1787-1)/9 + 5*(10^1786+1)

[PRIMO - Primality Certificate]
Version=2.1.1
WebSite=http://www.ellipsa.net/
Format=3
ID=B290C02E07C87
Created=07/08/2003 01:24:31 PM
TestCount=253
Status=Candidate certified prime

[Running Times]
Initialization=5.11s
1stPhase=12h 13mn 36s
2ndPhase=50mn 19s
Total=13h 4mn 1s

[Candidate]
File=C:\Primo IN & OUT files\PDP1787.in
Expression=2*(10^1787-1)/9+5*(10^1786+1)
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
HexadecimalSize=1484
DecimalSize=1787
BinarySize=5936

Proved prime with 'Primo 2.1.1'
by Patrick De Geest using a 3000 MHz Pentium 4 cpu.
Certificate Primo-B290C02E07C87-01.out available by
simple email request (337 KB).

2*(10^7277-1)/9 + 5*(10^7276+1)

Xinyao Chen informs me (May 19,2022) that this number is
now a proven palindromic plateau & depression prime.
Proof was done by Maksym Voznyy on November 30, 2020.

See source at
https://stdkmd.net/nrr/records.htm#pdprime

2*(10^19463-1)/9 + 5*(10^19462+1)

3-PRP!

2*(10^24215-1)/9 + 5*(10^24214+1)

3-PRP!
PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8].

2*(10^51779-1)/9 + 5*(10^51778+1)

PFGW Version 3.3.6.20100908.Win_Stable [GWNUM 25.14] 
Primality testing (65*10^51778+43)/9 [N-1/N+1, Brillhart-Lehmer-Selfridge] 
Running N-1 test using base 2 
Running N+1 test using discriminant 5, base 1+sqrt(5) 
(65*10^51778+43)/9 is Fermat and Lucas PRP! (1532.2989s+0.0028s)

2*(10^131393-1)/9 + 5*(10^131392+1)

By Tyler Busby dd. January 11, 2023.

Source : https://stdkmd.net/nrr/7/72227.htm#prime

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Patrick De Geest - Belgium

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