Problems & Puzzles: Conjectures

Conjecture 82.  Average of log Dn / log(logPn) equal R = 0,877 08...

Alain Rochelle sent the following conjecture, on March 17, 2019:

Denoting D(Pm) = P(n) - P(m) the difference between two consecutive primes Pm and Pn (with n=m+1) and denoting logX the natural logarithm of the number X, we consider the set of N consecutive primes included between X and X+X/(logX)^2.

The mathematic notation is :

X < P1 < P2 < ...... < PN-1 < PN < X+X/(logX)^2 < PN+1

For X to infinite, denoting Dn = Pn+1
 - Pn, we conjecture that the average of log Dn / log(logPn) equal R = 0,877 08... 

Denoting SIGMA[...] as the sum for n=1 to n=N, we write :

A(X) = (1/N) x SIGMA[ logDn / log(logPn) ] ---> R  (for X to infinite)

In practice, we verify by computer computation for increasingly values of X :MAG[A(X) - R] is of the order of log(X) / SR(X)    (with MAG[ ] the magnitude of [ ] and SR(X) the square root of X)

For example, I compute :

X = 2 x 10^8 ; N = 28 693 ; A(X) = 0,877 20

X = 4 x 10^8 ; N = 51 389 ; A(X) = 0,876 91

X = 6 x 10^8 ; N = 72 791 ; A(X) = 0,876 78

X = 8 x 10^8 ; N = 93 010 ; A(X) = 0,876 69

X = 9,8 x 10^8 ; N = 110 340 ; A(X) = 0,877 51 and logX/SR(X) = 0,000 66

The purpose consists to verify if this conjecture is true for large values of X :

X = 10^9 ; X = 2 x 10^9 ; X = 10^10 ; X = 10^11 ; X = 10^12  and so on ...

Q. Send your own verifications


 

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