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Conjecture: The sequence has infinitely many terms. In general, for any integers a,b,c with a>0 and gcd(a,b,c)=1, if b^2-4*a*c is not a square, a+b+c is odd, and gcd(b,a+c) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2 + b*prime(p) + c = prime(q).

Conjecture: The sequence has infinitely many terms. In general, for any integers a,b,c with a>0 and gcd(a,b,c) = 1, if b^2-4*a*c is not a square, a+b+c is odd, and gcd(b,a+c) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2 + b*prime(p) + c = prime(q).

Conjecture: The sequence has infinitely many terms. In general, if a,b,c are for any integers a,b,c with a>0 and gcd(a,b,c) = 1, and if b^2-4a4*a*c is not a square , a+b+c is odd, and gcd(b,a*+c+1) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2 + b*prime(p) + c = prime(q).

Conjecture: The sequence has infinitely many terms. In general, if a>0, ,b and ,c are integers with a>0 and gcd(a,b,c) = 1, and b^2-4a*c is not a square and gcd(b,a*c+1) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2 + b*prime(p) + c = prime(q).

Cf. A000040, A005574, A049002, A062326, A092057, A106483, A230351, A237413, A260120, A261281, A261339, A261352, A261361.

Conjecture: The sequence has infinitely many terms. Similarly, there In general, if a>0, b and c are infinitely many primes p integers with 2*primegcd(pa,b,c) = 1, and b^2 - 1 = prime(q) for some prime q, 4a*c is not a square and also gcd(b,a*c+1) is not divisible by 3, then there are infinitely many primes prime pairs {p with (,q} such that a*prime(p)-1)^2 + 1 = b*prime(qp) for some +c = prime (q).

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Cf. A000040, A005574, A049002, A062326, A092057, A106483, A230351, A237413, A261281, A261339, A261352, A261361.

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