**Srinivasa Ramanujan: Going Strong at 125**(

**Krishnaswami Alladi**, Editor - Notices of the AMS -

**Part I**and

**Part II**).

**The Legacy of Srinivasa Ramanujan-An International Conference**, University of Delhi, India

Solution to Problem B-1106 – Fibonacci Quarterly 50, Number 2 (May 2012)

Provided by the ONU-SOLVE Problem Group Ohio Northern University

Provided by the ONU-SOLVE Problem Group Ohio Northern University

On October 31, 2012, I was invited to speak in the **University of Findlay (OH) Mathematics Colloquium**. The title of my talk was **"***Beyond High School Science Fairs: The Senior Capstone Project*". The talk was an itinerary through a number of themes investigated in my undergraduate research projects (with an emphasis on quadratic residues and their applications to combinatorics, greater prime factor sequences and the relevant open problems in the area, etc). A difficult problem that I tried to address: what exactly should be considered as a sign of authenticity for a genuine undergraduate research experience, what would make it more than a science fair project (where the emphasis lies - granted with notable exceptions - on contingent data manipulation, with mathematics playing a secondary, albeit cool, "supporting role"). I suppose the answer has something to do with a serious dose of mathematical maturity reflected in a significant focus of action/intentionality
on abstract objects, in an increased ability of “bracketing out” the contingent
world, and in the sense of freedom implicit in this act.

A nice "Paley digraph" picture - with outgoing edges from *x* to *x+1, x+2, x+3, x+4, x+6, x+8, x+9, x+12, x+13, x+16 *and* x+18* (modulo 23) for *x = 0,1,...,22*.

An excellent article by Robert Gavin, in **"Academic Excellence - The role of research in the physical sciences at undergraduate institutions"** (Michael P. Doyle, Editor - published in the year 2000 by Research Corporation - a foundation for the advancement of science). Even if the paper, which emphasizes the role of publishing in a research-based education, refers to physical sciences, the ideas in there are even better suitable for mathematical sciences, where there is not an excessive need for laboratories and equipment.

Straight to the point:* "Publishing research articles, especially those done in collaboration with undergraduate students, should be expected, encouraged and supported both before and after the tenure decision".*

Straight to the point:

By Stuart Wolpert, October 04, 2012

Science 5 October 2012

(image source: http://en.wikipedia.org/wiki/File:Center_Milky_Way.jpg )

Photo source - wikipedia“The inner force that drives mathematicians isn’t to look for applications; it is to understand the structure and inner beauty of mathematics.”(source: New York Times)

We are no longer able to assert merely that "Grass in green!" Instead we must add something like:

A team of Harvard scientists has studied 9328 blades of grass from 37 randomly selected countries. They measured the wave length of light emanating from each blade when placed in the noonday sun in Harvard Yard. 98.32% produced light of wave length between 520 and 570 nanometers which is the accepted standard measure for green as certi ed by the International Bureau of Standards.

Proposed problem 1873

Mathematics Magazine, Vol. 84, No. 3, June 2011

Solution provided by the ONU-SOLVE problem group

Mathematics Magazine, Vol. 84, No. 3, June 2011

Solution provided by the ONU-SOLVE problem group

Proposed problem 1872
- Mathematics Magazine, Vol. 84, No. 3, June 2011

Solution provided by the ONU-SOLVE problem group.

Solution provided by the ONU-SOLVE problem group.

Camera shot of a telescope projection (courtesy of Ohio Northern University Observatory - public viewing, June 5, 2012, about 7:30 pm)

An exquisite musical interpretation!...

http://youtu.be/mzIy21pvWH8

*Un, deux, trois* - **Alexandrina Hristov**

http://youtu.be/mzIy21pvWH8

The sequence is:

**P** is the greatest prime factor function, then its infinite periodic extension **(X(n))** satisfies the recursion

*X(0)=X(1)=X(2)=p*), the prime sequences satisfying **(2)** ultimately enter the same limit cycle **(1)**. These are strange phenomena, of a similar nature to the **GPF-tribonacci sequences** (where we discovered that there are at least *four* distinct GPF-tribonacci limit cycles, of lengths 100, 212, 28 and 6). My last *MAPLE* experiment with **(2)** involved a seed consisting of three 80 digits primes,

**X(0) = 67525204474446805798439049565857966823399463795779492274451732592404979647691781**

**X(1) = 43280708928363322606959208124229795876921456250899031972019611375562107511756173**

**X(2) = 50558523494317177773504742464927157574684415344640904005922668959550436919047547**

which eventually led to the cyclic shift of**(1)** given by **7, 17, 3, 17, 5, 31, 5, 3, 5, 3, 3, 11, 13, 13, 37, 17, 11, 17, 7, 5, 5, 17**, with **7=X(50), 17=X(51), 3=X(52), ...**

**(1)** is the only one found. Today - improved the process of random selection of initial conditions. A sample in the last series of searches - still leading to **(1)** (log plot of the graph included below)

X(0)=38790912184195861716665094754005872283807697220069306337918513393335402427949730921338400505241

X(1)=34380504590591201002245563

X(2)=239197136656882070249024044251770704653065407229190712881313724790604984709544507179663

An interesting property of this sequence: if5, 3, 5, 3, 3, 11, 13, 13, 37, 17, 11, 17, 7, 5, 5, 17, 7, 17, 3, 17, 5, 31 (1)

The intriguing part: for virtually all random choices of the initial conditions that I made - that is, other than the trivial ones (leading to 1-cycles, such as those withX(n)=P[X(n-1)+3X(n-2)+2X(n-3)] (2)

which eventually led to the cyclic shift of

*Added on April 17, 2012*

X(0)=38790912184195861716665094754005872283807697220069306337918513393335402427949730921338400505241

X(1)=34380504590591201002245563

X(2)=239197136656882070249024044251770704653065407229190712881313724790604984709544507179663

Nikolai Petrovitch Bogdanov-Belsky (1868–1945)

Mental Calculation. In Public School of S. A. Rachinsky. Oil on canvas. The State Tretyakov Gallery, Moscow

Mental Calculation. In Public School of S. A. Rachinsky. Oil on canvas. The State Tretyakov Gallery, Moscow

"

George Andrews, the world's leading expert in the theory of integer partitions, discovered Ramanujan's Lost Notebook in 1976. In an interesting news story (

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