I just came up with a new mathematical constant. I don't know what its value is but here is how it works.Assume that there is always a prime number in the interval of [X, a*X] for X>1. The constant that I invented is the minimum of 'a'.By Bertrand's postulate it is proven that every 'a' equal or greater than two satisfies the condition. But can /sci/ work out what the minimum is? It shall be called the APIC constant (Anon's Prime Interval Constant).
1+epsilon. Infinity is so big you'll find a prime in that interval.
>>17031831For all X>1?Well, let us first look at [math]X=1+\varepsilon~(\varepsilon>0)[/math]. Naturally, [math]2\leq a(1+\varepsilon)[/math] because otherwise the interval contains no prime.Rearrange:[math]a\geq\frac2{1+\varepsilon} [/math]The smallest number that satisfies this is 2.But Bertand's postulate already gives the answer, the only thing that needed to be shown is that 2 is included in the range of valid a.
>>17031831You’d probably have better luck with something like [X, a(log X)^2].
>>17031831How does APIC stack up with the other constants (pi, e, zeta(2), etc.)?
