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Old 2017-11-28, 07:44   #1
carpetpool
 
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"Sam"
Nov 2016

23·41 Posts
Default Sieve for divisibility sequences

Prior to other existing sieving programs, it would be convenient to have a sieve for divisiblity sequences S(n) (prime n only) which include, but are not restricted to (a^n-b^n)/(a-b), LucasU(a,b,n), and LucasV(a,b,n).

The idea is that there are three arguments to the sieve, S(n) the sequence (already defined), the primes n in the range [a, b], and the sieve limit l, to which S(n) is trial divided by primes of the form 2*k*n+1 (or in some sequences trial divide by primes of the form 2*k*n+-1) with k <= l.

The sieve formats should be similar to that of other programs, such as in perl ntheory, msieve, srsieve, etc:

[S(n),a,b,l,1] to sieve S(n) for prime indices n, in the range [a,b] and divide by prime numbers of the form 2*k*n+1 up to k = l.

[S(n),a,b,l,-1] to sieve S(n) for prime indices n, in the range [a,b] and divide by prime numbers of the form 2*k*n+1 or k*n-1 up to k = l.

For instance, if one chooses to test S(n) = 2^n-1 for odd prime n up to 100, with l = 10000, the input is:

[2^n-1,3,100,10000,1]

and output are the primes:

[19, 31, 61, 67, 89]

because these primes are not divisible by any prime factor of the form 2*k*n+1 with k < 10000. One can write an ABC file (if there are several primes remaining in output) or test them manually if there are not many and or they are small values:

PFGW Version 3.8.3.64BIT.20161203.Win_Dev [GWNUM 28.6]

Primality testing 2^19-1 [N-1, Brillhart-Lehmer-Selfridge]
2^19-1 is prime! (0.0667s+0.1339s)
Primality testing 2^31-1 [N-1, Brillhart-Lehmer-Selfridge]
2^31-1 is prime! (0.1568s+0.1394s)
Primality testing 2^61-1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
2^61-1 is prime! (0.2173s+0.1997s)
Primality testing 2^67-1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Primality testing 2^89-1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
2^89-1 is prime! (0.3988s+0.1553s)

and find that 2^n-1 is prime for n = 2, 3, 5, 7, 11, 13, 17, 19, 31, 61, 89. The first 6 terms are a trivial result of dividing 2^n-1 by all prime factors of the form 2*k*n+1, with k > l, and 2*l*n+1 > 2^n-1. The others are a result of a primality test after 'sieving'. Other large sequences should work the same way, leaving less primes indices n to test in S(n).
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