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Martin Hopf
Martin Hopf
5:16 AM
GNFS complete on $C149 = \large \lfloor \pi^{298} \rfloor$ :

***factors found***

P49 = 2644539097105949684893162643607686362739000507267
P100 = 5349486191225615231548192678201994306809460610968311362443419264877753499269457696201291343378239179

ans = 1
 
 
2 hours later…
Peter
Peter
7:07 AM
F(19466)+F(19466+1)+F(19466+2)+F(19466+3)
F(35446)+F(35446+1)+F(35446+2)+F(35446+3)
F(36776)+F(36776+1)+F(36776+2)+F(36776+3)
F(44504)+F(44504+1)+F(44504+2)+F(44504+3)
 
 
3 hours later…
Peter
Peter
10:21 AM
F(44504)+F(44504+1)+F(44504+2)+F(44504+3)
F(51166)+F(51166+1)+F(51166+2)+F(51166+3)
F(56000)+F(56000+1)+F(56000+2)+F(56000+3)
 
Martin Hopf
Martin Hopf
$$F_n + F_{n+1} + F_{n+2} + F_{n+3} = F_{n+2} + F_{n+4} = L_{n+3}$$
The primes of the Lucas numbers are already in [OEIS](https://oeis.org/A001606).
 
Peter
Peter
Oops, I missed that this can be written as a SINGLE Lucas number. Well, errors are there to make them :)
 
Martin Hopf
Martin Hopf
Yes, that happens to me also :)
 
Peter
Peter
10:37 AM
Shall we try the next "pi-number" without a known factor ? The quadratic sieve will not help since it has $205$ digits.
$\lfloor \pi^{412} \rfloor$
 
Martin Hopf
Martin Hopf
GNFS on a 205 digit number would be a really challenge for NFS@Home. But hopeless for us.
 
Peter
Peter
I noticed that my expression is $L_{n+1}+L_{n+2}$ , but ... :)
Which expression could be of real interest ?
 
Martin Hopf
Martin Hopf
$$2 \cdot F_n + F_{n+2}$$

there are 40 primes for $n < 10 \ 000$.
 
Peter
Peter
And this is not in OEIS ?
 
Martin Hopf
Martin Hopf
$2 \cdot F_n + F_{n+2} = F_{n-2} + F_{n-1} + F_n + F_{n+2}$
right?
 
Peter
Peter
10:51 AM
Correct , or $F_n+L_{n+1}$
 
Martin Hopf
Martin Hopf
Ok
 
Peter
Peter
Before I invest PFGW-calculations again, we should make sure that this has not be done before by someone.
The above number factors into $$55677118588357803975816346699535953 (P35) \cdot P171$$
The first $27$ primes occur for $[2, 5, 6, 8, 9, 14, 18, 21, 24, 26, 33, 38, 54, 56, 78, 81, 90, 96, 98, 305, 414, 573, 689, 744, 816, 818, 969]$ , right ?
and PRP's oocur for
1389 1485 1998 2198 2366 2384 3041 4074 4866 6534 6774 6785 8006 11913
 
Martin Hopf
Martin Hopf
11:14 AM
Yes
41 primes so far.
 
Peter
Peter
We can also use factordb to approve this (or is that your source ?)
 
Martin Hopf
Martin Hopf
No I used pari
 
Peter
Peter
2*F(18896)+F(18896+2) seems to be the next
 
Martin Hopf
Martin Hopf
What is the syntax for Fibonacci numbers in factordb?
 
Peter
Peter
I(n) , I do not know it for Lucas-numbers, maybe L(n) ?
 
Martin Hopf
Martin Hopf
11:25 AM
Yes seems to be L(n)
 
Peter
Peter
2*F(24549)+F(24549+2)
2*F(26718)+F(26718+2)
2*F(27870)+F(27870+2)
 
Martin Hopf
Martin Hopf
Yes factordb confirms
 
Peter
Peter
Still less than 6k digits however
 
Martin Hopf
Martin Hopf
Maybe we should search for better sequences.
 
Peter
Peter
11:43 AM
What about $F_n^{L_n}+L_n^{F_n}$ ?
 
Peter
Peter
11:58 AM
Seems to be always even ...
2*F(41061)+F(41061+2)
 
 
1 hour later…
Peter
Peter
1:22 PM
2*F(60080)+F(60080+2)
 
 
3 hours later…
Martin Hopf
Martin Hopf
4:47 PM
Did you know about my 3D-simulation on 'Comets Hunting In Space' back in the year 2013?
 
Peter
Peter
5:07 PM
No , I did not. I passed $84\ 800$ without having found another PRP
 
Peter
Peter
5:42 PM
2*F(85290)+F(85290+2)
 

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