1. gcd(a_i, b_i)=1 for i=0,…,s-1;
2. we do not simultaneously have a_i = a_k and b_i = b_k for i ≠ k;
3. Let ω(p) be the number of distinct solutions of the congruence
   (a₀m+b₀)···(a_s-1m+b_s-1) ≡ 0 (mod p).
   Then ω(p) < p for all p ≤ s.

Then it was conjectured by L.E. Dickson in 1904 that a₀m+b₀,…,a_s-1m+b_s-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 a_im+b_i are simultaneously prime.

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