Problems & Puzzles: Conjectures Conjecture 9. Shank's Conjecture Defining : p_{n+1}= p_{n} + g(p_{n}) +1, then d_{n} =p_{n+1}  p_{n} = g(p_{n}) +1 (Ref.1, p. 250) The Shanks conjecture is the following : If p_{n}(g_{n}) is the first prime that follows a gap g_{n} between consecutive primes p_{n} and p_{n+1}, then Ln(p_{n}(g_{n})) ~ sqrt(g_{n}) ), which means that g_{n}/(Ln(p_{n}(g_{n}))^{2} ~ c. This c is in the range 0.70  0.82 (Ref. 2, p. 21) 




