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Zhi-Wei Sun, <a href="/A260753/a260753.txt">Checking the conjecture for s = -1, t = 1 and r = a/b with (a,b = 1..125)</a>
Conjecture: Any For any s and t in the set {1,-1}, every positive rational number r can be written as m/n with m and n in the set {k>0: prime(prime(k))-+s*prime(k)+1 t = prime(p) for some prime p}.
This conjecture implies that there are infinitely many primes p such that prime(p)-p+1 = prime(q) for some prime q.
We also have some other similar conjecturesIn the case s = -1 and t = 1, the conjecture implies that A261136 has infinitely many terms.
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We also have some other similar conjectures.
Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
Zhi-Wei Sun, <a href="/A260753/a260753.txt">Checking the conjecture for r = a/b with a,b = 1..125</a>
a(3) = 2279 since prime(prime(2279))-prime(2279)+1 = prime(20147)-20147+1 = 226553-20146 = 206407 = prime(18503) with 18503 prime, and prime(prime(2279*3))-prime(2279*3)+1 = prime(68777)-68777+1 = 865757-68776 = 796981 = prime(63737) with 63737 prime.