Problems & Puzzles: Conjectures Conjecture 6. Quantity of primes in a given range: Opperman, Brocard & Schinzel conjectures? The following conjecture was stablished by Opperman in 1882 : P(n^{2}+n) > P(n^{2}) > P(n^{2}n), (n>1). Which means that "between the square of a number and the square of the same number plus (or minus) that number, there is a prime" A close conjecture related with the above, is this one : "there is always a prime between x and x+(ln x)^2" Another close conjecture related with primes inside a range is the following due to Brocard, who in 1904 stated that : P(p^{2}_{n+1})  P(p^{2}_{n})=>4 for n=>2 which means that "between the squares of two consecutive primes greater than 2 there are at least four primes". (Ref. 1, p248) By his way, Schinzel conjectures that : for x>8, there is a prime between x and x+(lnx^{)2}. (Ref. 2, p. 7) 




