20160619, 15:25  #1 
"Mark"
Apr 2003
Between here and the
1110000101101_{2} Posts 
SmarandacheFibonacci Primes
I started a search for SmarandacheFibonacci primes. A SmarandacheFibonacci number is a concatenation of Fibonacci numbers. Using the notation SmF, one can specify a specific number in that sequence. For example, SmF_{6} = 11235813. So far only SmF_{2} and SmF_{4} are prime up to SmF_{1275}. I am currently sieving (with pixsieve) up to SmF_{7500} which is over 2 million decimal digits in length. This form is sieving very nicely. I have a little over 100 terms left and am still sieving.
One little "feature by accident" with pixsieve is that you can use a pfgw DECIMAL as input even when starting a new sieve. Because of this feature I was able to create an input file that only tested terms of specific lengths which corresponded to the SmF lengths rather than all lengths between two values. 
20160619, 17:20  #2 
"Robert Gerbicz"
Oct 2005
Hungary
2·19·43 Posts 
You can sieve this form much faster what your general string sieve does:
Code:
SmF_{k+1}=SmF_{k}*10^e(k+1)+Fib(k+1) if Fib(k+1) has e(k+1) decimal digits. Use that e(k+1) is e(k) or e(k)+1, so for fixed p prime: 10^e(k+1) mod p is just r=10^e(k) mod p or 10*r mod p. And obviously Fib(k+1)=Fib(k)+Fib(k1), here you now already Fib(k) and Fib(k1) mod p. So you need only one mulmod and other faster operations per each term. 
20160619, 18:23  #3  
"Forget I exist"
Jul 2009
Dartmouth NS
8450_{10} Posts 
Quote:


20160620, 16:02  #4  
"Mark"
Apr 2003
Between here and the
7,213 Posts 
Quote:


20160718, 12:49  #5 
"Mark"
Apr 2003
Between here and the
7,213 Posts 
Completed to SmF_{7500}. Considering how long these tests are starting to take (over 2 days on a fast i7) and how few are left after sieving, it will take quite a bit of luck to find another one. If anyone wants to try, you should write a custom sieve as described above. You will likely have well close to 1% of the candidates left to test after sieving.

20160718, 14:33  #6  
"Robert Gerbicz"
Oct 2005
Hungary
11001100010_{2} Posts 
Quote:


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