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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; and 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.


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