Least Squares Estimation of Density of Very Large Prime Numbers

• Activity Number: 483
• Type: Contributed
• Date/Time: Thursday, August 7, 2008 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Arthur Cohen*+
• Companies: Nassau Community College
• Keywords: least squares estimate ; density of prime numbers ; very large numbers ; evaluation of extrapolation ; Fundamental prime no. theorem ; Gauss' Logarithmic Integral of N

Abstract

Based on the actual number of prime numbers up to 10 to the 15th power, a least squares estimate is developed which relates the density of primes to the size of its associated order of magnitude interval. This estimate is exceedingly good. Since the actual data ends with a density of about 3%, I extrapolated to find that interval where the density is approximately 1%, and then .1%. The former came to a 43 digit number, and the latter came to a 390 digit number. These estimates were then compared to estimates generated using the Fundamental Prime Number Theorem as well as the first two terms of the series based on Gauss' Logarithmic Integral of N. These latter estimates agreed closely with my 1% estimate, but differed by about 10% with my .1% estimate; that is, they obtained a 435 digit number. Finally, for extremely large primes N, we show that the density approaches (1/ln N).