[QUOTE=Merfighters;236940]I think that should be 5↑5↑5↑5[SIZE=5]↑↑[/SIZE]0.421.[/QUOTE]

5↑↑0.421 = 5↑0.421

Either way is correct.

[QUOTE=3.14159;236905]To you, or to everyone?[/QUOTE]

Pretty much everyone, usually even including the authors of the paper introducing the new concept.

[QUOTE=Merfighters;236940]I think that should be 5↑5↑5↑5[SIZE=5]↑↑[/SIZE]0.421.[/QUOTE]

[QUOTE=3.14159;236944]5↑↑0.421 = 5↑0.421

Either way is correct.[/QUOTE]

In this case the definition is not complete!

I doubt there will be any extensions to the higher operators;

n↑↑↑x, n↑↑↑↑x, n↑↑↑↑↑x, n↑↑↑↑↑↑x etc..

Though there is the pentalog/pentaroot, hexalog/hexaroot, etc.

5↑↑↑1 = 5;

5↑↑↑2 = 5↑↑5 = 5↑5↑5↑5↑5 = 10↑10↑(1.3357*10↑2184).

5↑↑↑3 = 5↑↑5↑↑5 = 5↑↑(5↑5↑5↑5↑5) = 5↑↑(10↑10↑(1.3357*10↑2184)) = 5↑5↑5...↑5 (10↑10↑(1.3357*10↑2184) terms)

Etc, etc. There should be a real value for 5↑↑↑2.71, or 5↑↑↑(ln(2)). For now, it remains undefined for non-integers.

Alright.. A prime to be submitted;

479644*108[sup]13886[/sup]+1 (28242 digits)

Verification:

Primality testing 479644*108^13886+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using zero-padded FFT length 12K on 479644*108^13886+1
Running N-1 test using base 13
Special modular reduction using zero-padded FFT length 12K on 479644*108^13886+1
Calling Brillhart-Lehmer-Selfridge with factored part 70.38%
479644*108^13886+1 is prime! (44.5454s+0.0222s)

Submissions: 254692*54[sup]12680[/sup]+1 (21973 digits)
1124*2600!+1 (7760 digits)

Proofs;


Primality testing 254692*54^12680+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using zero-padded FFT length 10K on 254692*54^12680+1
Calling Brillhart-Lehmer-Selfridge with factored part 82.60%
254692*54^12680+1 is prime! (14.7984s+0.0015s)

Primality testing 1124*2600!+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2609
Generic modular reduction using generic reduction FFT length 2560 on A 25762-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 35.17%
1124*2600!+1 is prime! (4.3044s+0.0014s)

Submissions: 7203*(700!*p(700)#)^2+1 (7879 digits).

Verification:

Primality testing 7203*(700!*p(700)#)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5297
Generic modular reduction using generic reduction FFT length 2560 on A 26174-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 33.60%
7203*(700!*p(700)#)^2+1 is prime! (4.7043s+0.0013s)

1594^1594*797#+1 (5435 digits)

Primality testing 1594^1594*797#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 821
Calling Brillhart-Lehmer-Selfridge with factored part 85.16%
1594^1594*797#+1 is prime! (8.2406s+0.0142s)

[QUOTE=lorgix;240118]1594^1594*797#+1 (5435 digits)

Primality testing 1594^1594*797#+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 821
Calling Brillhart-Lehmer-Selfridge with factored part 85.16%
1594^1594*797#+1 is prime! (8.2406s+0.0142s)[/QUOTE]


Entry accepted.

4007#*4008+1 (1713 digits)

Primality testing 4007#*4008+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 4051
Calling Brillhart-Lehmer-Selfridge with factored part 33.36%
4007#*4008+1 is prime! (1.9568s+0.0022s)

The following should be all primes of the form p#(p+1)+1, where p is prime and <15000

[CODE]Primality testing 5#*6+1 [N-1, Brillhart-Lehmer-Selfridge]
small number, factored prime!
5#*6+1 is prime! (0.0578s+0.0558s)
Primality testing 13#*14+1 [N-1, Brillhart-Lehmer-Selfridge]
small number, factored prime!
13#*14+1 is prime! (0.0108s+0.0318s)
Primality testing 17#*18+1 [N-1, Brillhart-Lehmer-Selfridge]
small number, factored prime!
17#*18+1 is prime! (0.0292s+0.0459s)
Primality testing 19#*20+1 [N-1, Brillhart-Lehmer-Selfridge]
small number, factored prime!
19#*20+1 is prime! (0.0282s+0.0529s)
Primality testing 31#*32+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 37
Calling Brillhart-Lehmer-Selfridge with factored part 35.71%
31#*32+1 is prime! (0.3568s+0.0650s)
Primality testing 1399#*1400+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 1429
Calling Brillhart-Lehmer-Selfridge with factored part 33.49%
1399#*1400+1 is prime! (0.5338s+0.0448s)
Primality testing 1637#*1638+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 33.49%
1637#*1638+1 is prime! (0.4930s+0.0381s)
Primality testing 2131#*2132+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2141
Calling Brillhart-Lehmer-Selfridge with factored part 33.36%
2131#*2132+1 is prime! (0.9065s+0.0165s)
Primality testing 3457#*3458+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 33.53%
3457#*3458+1 is prime! (3.0264s+0.0662s)
Primality testing 4007#*4008+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 4051
Calling Brillhart-Lehmer-Selfridge with factored part 33.36%
4007#*4008+1 is prime! (3.7708s+0.0573s)
Primality testing 8179#*8180+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 8191
Calling Brillhart-Lehmer-Selfridge with factored part 33.39%
8179#*8180+1 is prime! (31.9143s+0.0230s)[/CODE]8179#*8180+1 has 3503 digits