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)

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