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• ## Re: Lucas Sequences. Why is S0=4?????

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• ... book I ... help!! ... Hi matt, Take a look at this paper : http://www.maths.tcd.ie/pub/ims/bull54/M5402.pdf You will understand that in fact, S0=4 is one
Message 1 of 5 , Dec 7 2:23 PM
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wrote:
>
> I have a quick question. How did Lucas determine s of 0 =4? The
book I
> am reading says he uses U of n and V of n for (P,Q). IF (P,Q)
> determines that s of 0 =4, Which equation did he use or which (P,Q)
> did he use? Or, is 4 used because it is the first composite number.
> Because if you use 2 or 3, you get prime numbers. Thanks for any
help!!
>
>
> -matt
>

Hi matt,

Take a look at this paper :
http://www.maths.tcd.ie/pub/ims/bull54/M5402.pdf

You will understand that in fact, S0=4 is one possibility but that
there are in fact infinitely many others (look at theorem 4 on page
71).

For example, S0=10 or S0=52 or S0=724 work as well.

Regards,

Jean-Louis.
• ... I have not heard of searches for large splitable primes. One way to search them is to concatenate known primes. 27918*10^10000+1 was found in 1999 by Yves
Message 2 of 5 , Dec 7 4:07 PM
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Ed Pegg Jr wrote:

> 2^17 - 1 = 131071. Both 131 and 071 are prime, or 131~071,
> or 3~(2^17-1), since it splits at the third position.
>
> What is the largest known splitable prime?

I have not heard of searches for large splitable primes.
One way to search them is to concatenate known primes.

27918*10^10000+1 was found in 1999 by Yves Gallot.
58789 is the smallest prime which can be put in front and give a prime:

D:>pfgw -t -q"5878927918*10^10000+1"
PFGW Version 1.2.0 for Windows [FFT v23.8]

Primality testing 5878927918*10^10000+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 69.83%
5878927918*10^10000+1 is prime! (26.4531s+0.0023s)

The form k*10^n+1 was chosen to make the concatenation easily provable.
PrimeForm/GW prp tested and APTreeSieve sieved.

--
Jens Kruse Andersen
• I found two more splitable primes, 127,031 (127)(031) and 839809 (839)(809). -matt [Non-text portions of this message have been removed]
Message 3 of 5 , Dec 7 7:03 PM
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I found two more splitable primes, 127,031 (127)(031) and 839809
(839)(809).

-matt

[Non-text portions of this message have been removed]
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