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(n2+n) > P(n2) > P(n2-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(p2n+1) - P(p2n)=>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)