If I may try to rephrase a question: Consider sequences S(p,q_0) = q_0, q_1, q_2, ..., where q_(n+1) = p*(q_n)+1. 1. Do there exist a prime p and number q_0, such that all elements of S(p,q_0) are prime? 2. For any M, do there exist a prime p and number q_0, such that the first M elements of S(p,q_0) are prime? (I think that's what Dan asks.) For (1), I suspect not. Consider any prime element q_k greater than p: Modulo q_k, the sequence is periodic, has 0 in the k'th position, and so will eventually have another zero. There, the original sequence has a multiple of q_k, but not q_k itself. A period of S modulo prime q_k is the order of p in q_k's multiplicative group. p and (p-1) are coprime to q_k because q_k>p. Let r=1/(p-1) mod q_k. Then the sequence S(p,q_0)+r (r+q_0, r+q_1, r+q_2, ...) is purely multiplicative: r+q_(n+1) = p*(r+q_n), and r+q_n = p^n*(r+q_0). So S+r (modulo q_k) has a period equal to p's order, and so does S, and its fundamental period is a factor of that. (Does anyone know which factor?) For (2), I can only speculate: q_0 limits the number of consecutive primes; but maybe we can beat any particular M, by choosing a sufficiently large q_0. -- Don Reble djr@nk.ca