### Finding prime constellations using linear polynomials

Suppose positive integers a0,…,as-1 and integers b0,…,bs-1 satisfy
1. gcd(ai, bi)=1 for i=0,…,s-1;
2. we do not simultaneously have ai = ak and bi = bk for i ≠ k;
3. Let ω(p) be the number of distinct solutions of the congruence
(a0m+b0)···(as-1m+bs-1) ≡ 0 (mod p).
Then ω(p) < p for all p ≤ s.
Then we call the sequence a0m+b0,…,as-1m+bs-1 an admissible sequence. See Primzahlverteilung, Karl Prachar, Satz 4.2, p. 45.

Then it was conjectured by L.E. Dickson in 1904 that a0m+b0,…,as-1m+bs-1 will simultaneously take on prime values for infinitely many m. See The New book of Prime Number Records, P. Ribenboim, p.372.

Examples

1. The admissible sequence m, m + 2 corresponds to the twin primes conjecture, i.e. where p and p+2 are primes infinitely often.
2. More generally, the admissible sequence m, m + 2k, k ≥ 1, corresponds to de Polignac's conjecture, i.e. where for fixed positive n, p and p+2k are primes infinitely often.
3. The admissible sequence m, 2m + 1 gives Sophie Germain primes p, i.e. where p and 2p+1 are primes infinitely often.
4. The admissible sequences m, m + 2, m + 6; m, m + 4, m + 6; m, m + 2, m + 6, m + 8; m, m + 4, m + 6, m + 10 were studied by Hardy and Littlewood in Some problems of ‘Partitio Numerorum’; III: on the expression of a number as a sum of primes, Acta Mathematica, 1923, Volume 44, Issue 1, pp 1-70 , especially p. 63, where there are errors in the tables.
5. Alan Offer has pointed out that the sequence im+n+1-i, 1 ≤ i ≤ n, n even, is admissible for n = 2, 4, 6, 10, 12, 16 and 18, but is admissible for n = 8, 14 and 20. When n = 10, apart from m = 1, we have not found another value of m for which all the aim+bi are simultaneously prime.
Our program first checks if the sequence of linear polynomials is admissible. If so, let M be the least m ≥ 1 such that all of the aim+bi are > 1. Then we find all values of m in the range u ≤ m ≤ v, where u ≥ M and for which all of a0m+b0,…,as-1m+bs-1 are certified to be primes under the Lucas - strong base 2 pseudoprime test. This is a BCMath version of a BC program. We restrict s to the range 2 ≤ s ≤ 6.

Enter the s positive integers ai separated by spaces:

Enter the s integers bi separated by spaces:

Enter u (≥ 1):
Enter v (≥ u):

Last modified 30th March 2014
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