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Associated Calculator Programs
Jim Cullen

Introduction


Over the years I've accumulated many pages of mathematical notes, problems, solutions, computer programs, calculator functions, etc. As time permits (and time is not so permitting lately) I will work to post more pages so that they may be more useful to someone than they would be stashed away in a drawer somewhere.

Just about anyone who knows me would be able to tell you that I have a 'thing' for π and for the Fibonacci Numbers. Now I have a formula that approximately connects the value of π to the value of the "Golden Ratio" φ. The value of φ may be calculated as (1+√5)/2. The approximation I've found in a computer search is accurate to twelve significant digits:

Pi-Phi formula: 821*Pi^5=5348*Phi^8 to 12 significant digits.

Probably my most important mathematical discovery to date is that of a Ramanujan-like series formula for 1/π4, the first of its kind, discovered using PSLQ integer relations software in a high-speed script I wrote for the Pari GP software package.

"Cullen's Pi Formula" discovered on Dec 5, 2010

Pi Formula discovered Dec 5, 2010



This formula has since appeared in several publications, among them: Tito Piezas' A Compilation of Ramanujan-Type Formulas for 1/πm; Jesus Guillera's Ramanujan-like Series and String Theory, p44; Wadim Zudilin's Arithmetic Hypergeometric Series, University of Newcastle, Callaghan, Australia, Feb2011, p33; Jonathan Borwein's Ramanujan and Pi, University of Newcastle, Apr2012, pp4,5; Jonathan Borwein's Meetings with Computer Algebra and Special Functions, also University of Newcastle, Oct2011, pp23,24; Jesus Guillera's Some Challenging Formulas for Pi, May2012, p2; Jesus Guillera's Kind of Proofs of Ramanujan-Like Series (PDF), Oct2012, p10; the December 2012 issue (Volume 59 Number 11) of the Notices of the American Mathematical Society, Srinivasa Ramanujan: Going Strong at 125, Part I, p1536; the interesting Pi Day article by David H. Bailey and Jonathan Borwein, Pi Day is upon us again and we still do not know if Pi is normal, University of Newcastle, May2013, p18; an article by David H. Bailey and Jonathan M. Borwein, Experimental Computation as an Ontological Game Changer, May2014, p11; and a mention in Pi: The Next Generation: A Sourcebook on the Recent History of Pi and Its Computation by David H. Bailey and Jonathan M. Borwein, July2016. This book, a compendium of 25 published papers, is a companion volume to Pi: A Source Book, by J.L. Breggren, Jonathan Borwein, and Peter Borwein, 2004 Third Edition. This classic work documents the history of Pi from ancient times to modern day.

I've been questioned several times over the years regarding the possible existence of other formulas of this type, involving even higher reciprocal powers of π. The short answer is, "Not even a hint of a possibility". The search algorithm I devised breaks the search area into many smaller regions, storing pre-computed values into memory for each region. This trade-off of memory for speed took some tweaking to optimize but it allowed me to find the π4 formula in about two weeks time, from conception of the original code to the discovery of the π4 formula itself. The algorithm can also proceed through the regions in successive sweeps, increasing in precision, to further speed the search for possible solutions. For about a year after this, I continued searching for higher-order solutions. I searched extensively for solutions involving π5 to π8, did some searching in the range of π9 to π12, and poked around sporadically from π13 to π16. No higher-order solutions have ever been found and I personally believe it likely that no further solutions will be found, at least not in any foreseeable future.


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