Anderson's Theorem (a) The number of primes in [1,n] is no more than 2+floor(n/2). The probability of n being prime when n is not prime is 1/2 - see Dasgupta,Papadimitriou,Vazirani "Algorithms" page 26. Therefore, the E(pi(n)) is n/2. (b) There does not exist another set of adjacent primes other than {1,2,3} 5: 2 + floor(5/2) = 2 + 2 = 4:=> {1,2,3,5} : 4 <= 4 7: 2 + floor(7/2) = 2 + 3 = 5 => {1,2,3,5,7} : 5 <= 5 11: 2 + floor(11/2) = 2 + 5 = 7 => {1,2,3,5,7,11} 6 <= 7 26: 2 + floor(26/2) = 15 => {1,2,3,5,7,11,13,17,19,23} : 10 <= 15 Lagrange's Theorem is Inaccurate Lagrange's theorem about primes states that pi(x) is the number of primes <= x. The pi(x) is approximately x/ln(x). He postulated that the lim of pi(x)/(x/lnx) as x-> infinity was 1. This is incorrect. if the number of primes is bounded by n/2 then refactoring and reducing Lagrange's Theorem results in the lim of ln(x) as x approaches infinity. This is alwa