Depression and Plateau Primes

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

*** VFYPR 1.13F  F_max=100000 S_min=100000 h=0 a=0 C=0 J=0 D=0 
N=(35*10^3-53)/99
N=353
*** N is prime!
Time: 0 sec

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

run"aprt-cle
Test number  N=? 355555553

Preparatory test
    Pass !

 355555553 is prime.
  0:00:00
OK

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

5*(10^141-1)/9 - 2*(10^140+1)

== ID:B28690376117A =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 355555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    555555555555555555553

Decimal size = 141
Binary size = 467


Started 01/26/2003 04:07:49 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.

5*(10^231-1)/9 - 2*(10^230+1)

== ID:B286A000468C0 =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 355555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555553

Decimal size = 231
Binary size = 766


Started 01/27/2003 12:04:48 AM
Running time 16s

Candidate certified prime

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

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

5*(10^427-1)/9 - 2*(10^426+1)

[PRIMO - Primality Certificate]
Version=2.0.0 - beta 3
Format=2
ID=B286B0165464E
Created=01/28/2003 06:30:14 AM
TestCount=74
Status=Candidate certified prime

[Candidate]
File=C:\Program Files\Primo200\pwp353_427.in
Expression=5*(10^427-1)/9 - 2*(10^(425+1)+1)
N$=1F5F714A5D852E5F79795F19A9173F558224BEF41086AC365B0016449A443E3E8CA2DDD70EEA1C8670AAF714A0915D3905399F2663AD523FCB830929F5353459C3784E259272468A0A962587BA47C9F87495DDF1844ABB5F3F87E1120FDD5FD3033A2929459B4AF454E4DB82BAF3CF009475396B99CB23EC0C9C55C0E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E38E1
HexadecimalSize=355
DecimalSize=427
BinarySize=1417

[Running Times]
Initialization=0.36s
1stPhase=1mn 38s
2ndPhase=28.50s
Total=2mn 7s

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

5*(10^463-1)/9 - 2*(10^462+1)

== ID:B286C01100300 =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 355555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    555555555555555555555555555555555555555555555555555555555555\
    5555555555555555555555555555555555555555553

Decimal size = 463
Binary size = 1537


Started 01/29/2003 04:57:06 AM
Running time 6mn 52s

Candidate certified prime

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

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

5*(10^727-1)/9 - 2*(10^726+1)

[PRIMO - Primality Certificate]
Version=2.0.0 - beta 3
Format=2
ID=B286E00DB01D2
Created=01/31/2003 03:59:12 AM
TestCount=129
Status=Candidate certified prime

[Candidate]
File=C:\Program Files\Primo200\pdp353_727.in
Expression=5*(10^727-1)/9 - 2*(10^(725+1)+1)
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
HexadecimalSize=604
DecimalSize=727
BinarySize=2414

[Running Times]
Initialization=1.73s
1stPhase=2h 10mn 2s
2ndPhase=10mn 9s
Total=2h 20mn 14s

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

5*(10^1975-1)/9 - 2*(10^1974+1)


Also 32*(10^1974-1)9+1.
was the largest proven plateau prime known in May 2001. In base 10,
it consists of 1973 successive 5's sandwiched by a pair of 3's.

C:\Pdp>pfgw -hhd1974.fac -l -tm -q"32*(10^1974-1)/9+1"
PFGW Version 1.1 for Windows

Output logging to file pfgw.out
Primality testing 32*(10^1974-1)/9+1 [N-1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file hd1974.fac
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 33.36%
32*(10^1974-1)/9+1 is prime! (6.430000 seconds)

Helper file and proof by David Broadhurst.
Timing : reproduced on a Pentium III 650 MHz chip.


Compare the timings with this newly created Primo Certificate !

[PRIMO - Primality Certificate]
Version=2.1.1
WebSite=http://www.ellipsa.net/
Format=3
ID=B290D009E7A50
Created=07/09/2003 02:53:12 AM
TestCount=274
Status=Candidate certified prime

[Running Times]
Initialization=6.51s
1stPhase=21h 5mn 58s
2ndPhase=2h 7mn 17s
 Total=23h 13mn 22s

[Candidate]
File=C:\Primo IN & OUT files\PDP1975.in
Expression=5*(10^1975-1)/9-2*(10^1974+1)
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
HexadecimalSize=1640
DecimalSize=1975
BinarySize=6560

Proved prime with 'Primo 2.1.1'
by Patrick De Geest using a 3000 MHz Pentium 4 cpu.
Certificate Primo-B290D009E7A50-01.out available by
simple email request (414 KB).

5*(10^7231-1)/9 - 2*(10^7230+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 23, 2020.

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

5*(10^45861-1)/9 - 2*(10^45860+1)

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

5*(10^47305-1)/9 - 2*(10^47304+1)

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

5*(10^283827-1)/9 - 2*(10^283826+1)

by Serge Batalov dd. January 17, 2023.

Source : https://stdkmd.net/nrr/3/35553.htm#prime

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Patrick De Geest - Belgium

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