Abstract
The paper contains a geometric interpretation of Andrica’s conjecture about the gap between the square roots of the consecutive primes and brings empirical evidence that the random fluctuations of the gap between the quare roots of the consecutive primes seem to stabilize around the mean gap.
Highlights
In 1886, the German mathematician Leopold Kronecker attended the reunion of natural scientists in Berlin and said: “God made the integers; all else is the work of man.” The famous quote, echoing Pythagoras’ opinion, was cited by H
As of 2008, the inequality (5) has been mentioned to hold true for n up to 1.302 ×1016, but it is still a conjecture, based on the empirical observations of the tables with prime numbers. It gives an empirical upper bound for the gap between the square roots of the consecutive prime numbers but tells us noting about how these gaps behave
Dealing with the gaps between the square roots of the consecutive prime numbers, we can see a striking statistical stability, which cannot be detected when we look at the gaps between the corresponding consecutive prime numbers
Summary
In 1886, the German mathematician Leopold Kronecker attended the reunion of natural scientists in Berlin and said: “God made the integers; all else is the work of man.” The famous quote, echoing Pythagoras’ opinion, was cited by H. As mentioned on page 163 of the book [3], “Prime numbers have always fascinated mathematicians. They appear among the integers seemingly at random, and yet not quite. As more and more primes satisfy a certain conjecture about them, its credibility increases but as large the number 1018 is, it is still a finite number, whereas the set of prime numbers is infinite. The conjectures involving primes are useful and some are very ingenious but what is really needed is to get some statements and properties satisfied by all prime numbers, or at least to detect some kind of stability in the obvious random behavior of the primes
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