(Adrien-Marie Legendre) There is always a prime number between a square number and the next. To put it algebraically, given an integer , there is always a prime such that . Put yet another way, , where is the prime counting function.
This conjecture was considered unprovable when it was listed in Landau’s problems in 1912. Almost a hundred years later, the conjecture remains unproven even as similar conjectures (such as Bertrand’s postulate) have been proven.
But progress has been made. Chen Jingrun proved a slightly weaker version of the conjecture: there is either a prime or a semiprime (where is a prime unequal to ). Thanks to computers, brute force searches have shown that the conjecture holds true as high as .
|Date of creation||2013-03-22 16:38:17|
|Last modified on||2013-03-22 16:38:17|
|Last modified by||PrimeFan (13766)|