The Golden Ratio

The Golden Ratio Linked to e and the Prime Numbers

Barry Dalgarno

18 December 2009

    This is a short communication related to the discovery of a new law of prime numbers, based on finely-tuned geometric entanglement of the prime intervals, and an infinite structure of sequential triangular symmetry (Ref. 1). 

    The structure of minimum vertices also links the prime numbers and e (the base of natural logarithms) to phi, the Golden Ratio (Ref. 2). A graph is plotted, of cumulative average geometric entanglement (xi mean) versus the natural logarithm (base e) of cumulative average prime-count (log (NT mean)) (relating to completed triangles with respect to the structure of minimum vertices). Eventually the graph begins to settle down, revealing the overall trend. Now, a right-angled triangle constructed from the three points that correspond to xi (mean), log (NT mean) and the origin (zero), displays an interesting property. As NT increases, the ratio of the opposite side (xi mean) to the adjacent side (log (NT mean)) gradually converges on 1/2. The ratio of the hypotenuse to the adjacent side (log (NT mean)) converges on phi minus 1/2. This additional discovery is of potential interest in the wider field of mathematics, beyond that of number theory. We obtain the following relationship (the Golden Ratio is the symbol on the right).  

In the limit when NT approaches infinity, the formula is : 

Bibliography

Livio, M. (2002).  The Golden Section: the Story of Phi. Broadway Books, New York.

Walser, H. (2001). The Golden Section. The Mathematical Association of America, Washington.

Maor. E. (1994). e: The Story of a Number. Princeton University Press, Princeton, N.J.

Schimmel, A. (1993). The Mystery of Numbers. Oxford University Press, New York.

Stewart, I. (1995). Nature's Numbers. Weidenfeld and Nicolson, London.

References

1. Dalgarno, B. (2009). A New Law of Prime Numbers Based on an Infinite Structure of Sequential Triangular Symmetry from Finely-Tuned Geometric Entanglement of the Prime Intervals. (Initial communication).  http://sites.google.com/site/geometryoftheprimes/

2. Dalgarno, B. The Golden Ratio Linked to e and the Prime Numbers. (In preparation).