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### Harlan Brothers

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• Physics -3

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I particularly like the Witten quote, which has a timeless quality: "The field is still developing very fast, and it is difficult to guess what will happen or what the lessons will turn out to be."

### Harlan Brothers

Shared publicly -#PiDay

### Harlan Brothers

Shared publicly -*e*. Thanks, Richard! A short proof of the result can be found here:

http://www.brotherstechnology.com/docs/Finding_e_in_Pascal's_Triangle.pdf

A more detailed treatment that includes an interesting connection to the Barnes G-Function appears in

*The*

*Mathematical*

*Gazette*, Vol. 96, No. 535, 2012; pages 145-148.

#mathematics

**Pascal's triangle and the number e**

*Pascal's triangle*is a triangular array of integers, the first few rows of which are shown in the picture. Apart from the flanking strips of 1s, each entry is the sum of the two entries above it. There are many well known patterns among the entries of Pascal's triangle. Two of these patterns are (a) that the sum of the entries in the n-th row is equal to 2^n, and (b) that the alternating sum of entries in the n-th row is equal to 0. For example, if n = 4, this means (a) that 1+4+6+4+1 = 2^4 = 16, and (b) that 1-4+6-4+1 = 0.

A less studied sequence associated with Pascal's triangle is the

*product*of the entries in each row. This sequence, which appears as A001142 in

*The On-Line Encyclopedia of Integer Sequences*(http://oeis.org/A001142) begins

1, 1, 2, 9, 96, 2500, 162000, 26471025, ...

Let us denote the entries of this sequence by p(1), p(2), p(3), and so on. The remarkable property of the sequence, illustrated in the picture, is that as n becomes large, the value of p(n-1)p(n+1)/(p(n)^2) converges, albeit slowly, to the number e. The number e, which is approximately 2.718281828459045, is a famous irrational number, and is important because it is the base of the natural logarithm.

This post is based on a post by +Richard Elwes, which you can find here (https://plus.google.com/114216686796425894192/posts/EvZyEiKRJtn). The result itself is due to +Harlan Brothers. +Alexander Bogomolny has written up a nice proof of the result here (http://www.cut-the-knot.org/arithmetic/algebra/HarlanBrothers.shtml).

#mathematics

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### Harlan Brothers

Shared publicly -### Harlan Brothers

Shared publicly -- Brothers Technology, LLCFounder, 1992 - presentBrothers Technology creates a broad range of intellectual property for license and sale. Current projects range from commercial encryption techniques to novel consumer devices and educational tools.
- The Country SchoolDirector of Technology, 2002 - 2011Greatly advanced the use of technology through infrastructure upgrades, software training, and installation of interactive educational systems. Other roles included teacher, school administrator, advisor, and network administrator.
- Yale Mathematics DepartmentConsultant, 2001 - 2007- Edited the planned 20th anniversary edition of The Fractal Geometry of Nature for Benoit Mandelbrot - Collaborated with Michael Frame and Benoit Mandelbrot at the NSF funded Yale University Fractal Geometry Workshops - Conducted workshop and lab on the subject of fractal music - Trained educators in the subject of fractal geometry and in the use of specialized software - Developed fractal-based mathematics curricula for students in middle school through college

I'm an inventor, composer, mathematician, educator, and founder of Brothers Technology, LLC. I served as Director of Technology at The Country School for nine years. I've got six U.S. patents and have published numerous research papers related to number theory and fractal geometry. I enjoy performing and composing in many styles including jazz, pop, and Brazilian music.

Over the course of six years, I had the honor and the pleasure of working with Michael Frame and Benoit Mandelbrot at the NSF funded Yale University Fractal Geometry Workshops. The workshops served to train educators in the subject of fractal geometry with the goal of developing fractal-based mathematics curricula for students in middle school through college.

Latest projects include a provably secure, patent pending encryption system, an automatic interior mailbox light, and an interactive, multimedia birthday app for iPhone called iKandl.

**If you're in my circles and have an iPhone or iPad, let me know and I'll send you a free upgrade code for the eco-friendly iKand Birthday App! It ain't Angry Birds, but it *IS* fun and has the advantage of making other people smile!**

- Berklee College of MusicComposition
- Gateway Community CollegeMathematics

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