```

8*(10^3-1)/9 - (10^2+1)

*** VFYPR 1.13F  F_max=100000 S_min=100000 h=0 a=0 C=0 J=0 D=0
N=(78*10^3-87)/99
N=787
*** N is prime!
Time: 0 sec

8*(10^5-1)/9 - (10^4+1)

run"aprt-cle
Test number  N=? 78887

Preparatory test
Pass !

78887 is prime.
0:00:00
OK

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

8*(10^87-1)/9 - (10^86+1)

== ID:B28690085E1B6 =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 788888888888888888888888888888888888888888888888888888888888\
888888888888888888888888887

Decimal size = 87
Binary size = 289

Started 01/26/2003 02:26:14 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.

8*(10^113-1)/9 - (10^112+1)

== ID:B28690351CC48 =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 788888888888888888888888888888888888888888888888888888888888\
88888888888888888888888888888888888888888888888888887

Decimal size = 113
Binary size = 376

Started 01/26/2003 03:28:12 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.

8*(10^171-1)/9 - (10^170+1)

== ID:B286904009F20 =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 788888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888887

Decimal size = 171
Binary size = 568

Started 01/26/2003 06:39:09 PM
Running time 5s

Candidate certified prime

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

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

8*(10^567-1)/9 - (10^566+1)

== ID:B286C03C47DCE =============================================

PRIMO 1.2.1 - Primality Certificate

-----------------------------------------------------------------
Candidate
-----------------------------------------------------------------
N = 788888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888888888888888888888888888888888888888\
888888888888888888888888887

Decimal size = 567
Binary size = 1884

Started 01/29/2003 05:33:28 PM
Running time 12mn 28s

Candidate certified prime

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

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

8*(10^1689-1)/9 - (10^1688+1)

[PRIMO - Primality Certificate]
Version=2.1.1
WebSite=http://www.ellipsa.net/
Format=3
ID=B290B00657C49
Created=07/07/2003 01:50:55 AM
TestCount=259
Status=Candidate certified prime

[Running Times]
Initialization=4.46s
1stPhase=11h 10mn 1s
2ndPhase=1h 25mn 43s
Total=12h 35mn 48s

[Candidate]
File=C:\Primo IN & OUT files\PDP1689.in
Expression=8*(10^1689-1)/9-(10^1688+1)
DecimalSize=1689
BinarySize=5611

Proved prime with 'Primo 2.1.1'
by Patrick De Geest using a 3000 MHz Pentium 4 cpu.
Certificate Primo-B290B00657C49-01.out available by
simple email request (332 KB).

8*(10^8903-1)/9 - (10^8902+1)

3-PRP!

8*(10^115811-1)/9 - (10^115810+1)

Tested by Ray Chandler

PFGW Version 3.4.8.64BIT.20110617.Win_Dev [GWNUM 26.6]

Primality testing (71*10^115810-17)/9 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Generic modular reduction using generic reduction Core2 type-1 FFT length 48K, Pass1=64, Pass2=768 on A 384719-bit number
Running N-1 test using base 7
Generic modular reduction using generic reduction Core2 type-1 FFT length 48K, Pass1=64, Pass2=768 on A 384719-bit number
Running N+1 test using discriminant 17, base 1+sqrt(17)
Generic modular reduction using generic reduction Core2 type-1 FFT length 48K, Pass1=64, Pass2=768 on A 384719-bit number
Calling N-1 BLS with factored part 0.01% and helper 0.01% (0.04% proof)
(71*10^115810-17)/9 is Fermat and Lucas PRP! (5003.3912s+0.0088s)

```