Depression and Plateau Primes

2*(10^5-1)/3 - 5*(10^4+1)

run"aprt-cle
Test number  N=? 16661

Preparatory test
    Pass !

 16661 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*(10^13-1)/3 - 5*(10^12+1)

run"aprt-cle
Test number  N=? 2*(10^13-1)\3 - 5*(10^12+1)

Preparatory test
    Pass !

Main test  for P= 2
   for Q= 3    for Q= 5    for Q= 7    for Q= 13    for Q= 11

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

Main test  for P= 5
   for Q= 11     Pass !

 1666666666661 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*(10^17-1)/3 - 5*(10^16+1)

== ID:B286400BEAF3C =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 16666666666666661

Decimal size = 17
Binary size = 54

-----------------------------------------------------------------
1) N+1 Test
-----------------------------------------------------------------
N = Candidate
S = 6
R = 2777777777777777
Q = 3

-----------------------------------------------------------------
2) N+1 Test
-----------------------------------------------------------------
N = R of preceding test
S = 124686
R = 22278185023
Q = 7

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

Started 01/21/2003 03:28:16 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*(10^19-1)/3 - 5*(10^18+1)

== ID:B286400C0B44E =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 1666666666666666661

Decimal size = 19
Binary size = 61

-----------------------------------------------------------------
1) N+1 Test
-----------------------------------------------------------------
N = Candidate
S = 2094
R = 795924864692773
Q = 2

-----------------------------------------------------------------
2) EC Test
-----------------------------------------------------------------
N = R of preceding test
S = 6122937
R = 129990691
A = 0
B = 4
T = 2

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

Started 01/21/2003 03:30:29 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*(10^37-1)/3 - 5*(10^36+1)

== ID:B2865010BF94A =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 1666666666666666666666666666666666661

Decimal size = 37
Binary size = 121

-----------------------------------------------------------------
1) EC Test
-----------------------------------------------------------------
N = Candidate
S = 107511755169656392
R = 15502180799082133901
A =-1
B = 0
T = 2

-----------------------------------------------------------------
2) N+1 Test
-----------------------------------------------------------------
N = R of preceding test
S = 35946858
R = 431252734219
Q = 2

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

Started 01/22/2003 04:52:41 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*(10^53-1)/3 - 5*(10^52+1)

== ID:B2868048CA95C =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 16666666666666666666666666666666666666666666666666661

Decimal size = 53
Binary size = 174


Started 01/25/2003 09:12:07 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.

2*(10^73-1)/3 - 5*(10^72+1)

== ID:B2869006E9E5C =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 166666666666666666666666666666666666666666666666666666666666\
    6666666666661

Decimal size = 73
Binary size = 240


Started 01/26/2003 02:00:49 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*(10^101-1)/3 - 5*(10^100+1)

== ID:B286900941F60 =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 166666666666666666666666666666666666666666666666666666666666\
    66666666666666666666666666666666666666661

Decimal size = 101
Binary size = 333


Started 01/26/2003 02:41:47 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*(10^6233-1)/3 - 5*(10^6232+1)

Xinyao Chen informs me (May 19,2022) that this number is
now a proven palindromic plateau & depression prime.
Proof was done by Masaki UKAI on September 16, 2020.

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

2*(10^24029-1)/3 - 5*(10^24028+1)

3-PRP!
PFGW Version 20040816.Win_Stable (v1.2 RC1a) [FFT v23.8]

2*(10^40223-1)/3 - 5*(10^40222+1)

3-PRP!
PFGW Version 20040816.Win_Stable (v1.2 RC1a) [FFT v23.8]

2*(10^66395-1)/3 - 5*(10^66394+1)

By Serge Batalov dd. May 15, 2009.

(5*10^66394-17)/3 is 3-PRP! (217.2589s+0.0029s)
(5*10^66394-17)/3 is 23-PRP! (286.7449s+0.0033s)
 
Re-confirmation with -tc:
PFGW Version 1.2.0 for Windows [FFT v23.8]

Primality testing (5*10^66394-17)/3 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N+1 test using discriminant 11, base 1+sqrt(11)
Running N+1 test using discriminant 11, base 3+sqrt(11)
Calling N-1 BLS with factored part 0.01% and helper 0.00% (0.02% proof)
(5*10^66394-17)/3 is Fermat and Lucas PRP! (2667.8445s+0.0060s)
Done.

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Patrick De Geest - Belgium

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