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)