**Joseph
Arkin**

**Biography
by David C. Arney (USMA)**

Joseph
Arkin joined the Mathematics Department at West Point in 1986 "to do
math." Joe arrived at the
Academy with his wife Judy. Together
they had raised a family of four girls, Helen, Jessica, Aviva, and Sara.
There was no doubt that mathematics was and always will be his true
passion. He was deeply in love with
the subject. Yet, Joe wasn't
a typical mathematician. He was a
self-educated, amateur mathematician, completely dedicated to "doing"
mathematics.

Joe
Arkin learned his high-school mathematics at Brooklyn's Boys High.
He understood algebra and geometry, but had not studied trigonometry,
calculus, linear algebra, or analysis, although he definitely knew methods of
proof and logic. On the other hand,
he had taught himself methods of inquiry very well.
There was absolutely no mathematics problem that he wasn't deeply
interested in. He had great
intuition and tremendous feel for his forte, number theory.
He had a new idea everyday, a new problem to investigate every week, a
new direction to pursue every month. By
1986, Joe had already published 25-30 articles in prestigious journals and had
worked with several prominent mathematicians.

Joe
had a driving desire to select West Point for his mathematical home.
For several years, he had been a mathematical independent, with an
affiliation with the New York Academy of Sciences, and partnerships with several
mathematicians. His desire to work
at the Military Academy came from his experience as a soldier in the Army during
World War II. To Joe, doing
mathematics at West Point was like serving his country as he had done 40 years
before. Joe and Judy made the
25-mile, 35-minute car trip from Spring Valley to West Point, hundreds of times
over the next 10-12 years.

His
first mathematical inquiry at West Point was tiling the plane using rectangles
of sides with lengths of Fibonacci numbers.
His work on that problem eventually led to several papers and
presentations. Joe didn't know all
about higher mathematics, but he sure knew numbers. To Joe, every number was special, every pair of numbers
shared some properties, and every set of numbers were connected, somehow,
someway. Tiling using Fibonacci
numbers was a nice geometrical use of a sequence of numbers with special
properties. After that project, Joe
and his West Point colleagues took on the challenge of arranging numbers in 2-
and 3-dimensional arrays, squares and cubes, with special (called magical)
properties. Joe Arkin was a master
at this kind of number theory. At
first, Joe was not a fan of computers. In
his mind, they were hindrances to "real" mathematical thinking.
Computers couldn't do Joe's mathematics of conjecturing, analyzing, and
proving. Joe had a gift for finding
patterns. In Joe's mind, numbers
came alive, and he visualized the patterns and properties that were hidden to
others. Eventually, Joe began to
use computers to produce numbers that came from exhaustive iteration or
complicated formulas. Joe was
always in need of large prime numbers, and the computer was able to produce what
he wanted very quickly. Eventually,
theoretical mathematician Joe Arkin became a serious computer user.
Joe's West Point number theory team constructed all sorts of magic
squares and cubes with special properties.
One special cube was called the Cameron Cube, named for the department
head at West Point. Another one was
named the Supercube.

Joe
knew everyone in the number theory community.
He never forgot the people that he met.
He could tell stories about people that he had met only once at a
conference 20 years before. He
remembered the significant parts of their mathematics.
He just enjoyed meeting and talking with and about mathematicians.
He was kind, generous, and concerned about other people.
He loved people and, in return, everyone loved Joe.

Joe's
mathematical career started in the early 1960s. Joe was doing mathematics on his own and wanted to meet other
mathematicians to share ideas. He
had written a 30-page paper on a topic related to power series and their
properties. After some connections
were made, Joe visited F. A. Ficken at New York University.
Ficken gave Joe's paper to Richard Pollack, who suggested that Joe
rewrite the paper, making it shorter and more concise.
Pollack suggested sending the paper to Leonard Carlitz, a number theorist
at Duke. Under Carlitz's guidance, the paper was reworked, reduced to
2 pages, and published in the *American
Mathematical Monthly*.

The
Arkin family began a tradition of driving to the summer meetings of the American
Mathematics Society, no matter where they were in America -- San Jose, Vancover,
Columbus, Laramie, Missoula, Phoenix. A couple trips were taken across the entire country in the
late 1960s and early 1970s with a 17-foot trailer.
The Arkins were sightseers, who enjoyed their family vacation; and
travelers, trying to arrive at a professional mathematics meeting to present
state-of-the art research. Joe and
Judy have many wonderful memories of these cross-country excursions.

Joe's
first encounter with his close friend Paul Erdos came on a plane going to a
number theory conference at Washington State University in 1971.
This was Joe's first conference. Erdos
was giving one of the principal addresses. Joe was one of 18 others, including
his future friend and mentor Ernst Straus, invited to talk on their work.
Paul Erdos and Joe were seated nearby on the plane, and Joe asked Erdos
to read his paper entitled "Researches on Some Classical Problems." It was the start of their life-long friendship.
Joe also had a productive and rewarding relationship with Ernst Straus.
Joe met Straus through Paul Erdos while attending that conference. An accomplished mathematician, Straus, had worked with
Einstein and was a number theorist. Joe
Arkin and Straus became productive partners.
While they are listed as co-authors on only two papers and Joe extended
some of his work with Straus in a third paper, they were close collaborators and
friends for a number of years.

Arkin,
Straus, and Erdos were a strong trio of number theorists.
Straus was Einstein's assistant at the Institute for Advanced Study in
Princeton. Erdos was simply the
greatest mathematician of the 20^{th} century and the most prolific
author of all time. The three men
were colleagues when it came to thinking about number theory and doing math.
Each gaining ideas and inspiration from the others.
Mrs. Louise Straus, Ernst's wife, recalled this trio's relationship much
later when she wrote to Arkin: "Both Erdos and Ernst were very supportive
of your work, very encouraged and interested in your ideas.
I know this for a fact because I often heard them talking about you and
your work. If you were to mention
the encouragement you received from both of them, you would be doing them a true
honor." Joe's special
relationship with his colleagues was recounted in a eulogy for Straus given by
UCLA mathematician Albert Whiteman: "Ernst Straus was an avid collaborator.
He wrote numerous joint papers and I now wish to speak very briefly about
his work with two of his principal collaborators -- Paul Erdos and Joseph Arkin.
Professor Paul Erdos is one of the world's most famous mathematicians, a
grand master of mathematics. Erdos'
visits to UCLA were marked by intense discussions between Ernst and Paul.
The resulting interplay of ideas was both exhausting and highly
productive. A joint paper was
usually the outcome. It was my
privilege to listen in on some of the discussions between these two
mathematicians. It was awe-inspiring to hear them develop intricate arguments
without putting pencil to paper or chalk to blackboard.
Joseph Arkin, a disabled World War II veteran, was an amateur
mathematician. Unlike Erdos, he
seldom traveled. His joint work
with Straus was carried out mostly by correspondence and long distance telephone
conversations. Arkin is a highly
original and imaginative mathematician, but he has great difficulty in
expressing his ideas clearly and precisely.
Ernst Straus gave him friendship, guidance and encouragement.
Their collaborative efforts produced several beautiful papers on Latin
Systems and Diophantine equations."

In
the late 1960s, Joe Arkin was given an invitation to come to the Institute of
Advanced Study (IAS) at Princeton. The
only way real progress could be made was for Joe to move to Princeton and put
his entire effort into working with several expert collaborators.
It was a tempting offer and a great honor for Joe to be a scholar at the
IAS. However, it never happened.
Joe and Judy decided that such a change in lifestyle wasn't best for the
family. A compromise was designed
so that Joe would come down to Princeton once per week to spend the day and
contribute what he could. Joe has
great memories of that special year.

A
working group that was a natural for Joe Arkin was the Fibonacci Society.
Joe's first paper in the *Fibonacci
Quarterly* was published in 1965. Joe
met Vern Hoggatt through the Society and soon the two number theorists were
producing results and publishing papers together.
Joe attended the Fibonacci Society's international meetings.
The most recent meetings he attended were in Winston-Salem and Scotland.

Joe
Arkin desired to edit and publish a mathematics journal.
Joe wrote a pamphlet in May 1991 entitled "Senior Lecturer's Problem
Notebook." It's only issue contained an example problem,
"Archimedes' Trisection of an Angle," explained by Joe; no elementary
problems; and 4 advanced problems, 2 from Joe in number theory, and 2 from Paul
Erdos involving properties of sequences. Joe
was persistent and his publishing suggestions finally found an acceptable venue.
He came up with an idea for a newsletter that would publish articles of
interest to the mathematics departments of the service academies.
Much of the newsletter would be for sharing teaching and curricular
ideas, but there would be room for an occasional research article and
description of research programs. *Mathematica Militaris* was born with Joseph Arkin as its Founding
Editor. As evidenced by this issue,
eight years and 25 issues later, it still going strong.

Joe
finally had the opportunity to collaborate and co-author an article with Paul
Erdos. Most of the work on this
paper was done by mail. The
manuscript was mailed back and forth a few times.
On Aug 25, 1994, Erdos wrote back to Arkin:
"I am a few days in the hills near Budapest in a hotel of the
Academy; I was in Zurich 2 weeks ago. As
far as I can tell, your paper with Colonel Arney is new and interesting; can you
extend it for prime triples, p, p+m1, p+m2."
Later Erdos approved the paper title and journal submission.
After its publication in 1996, Joe Arkin held the honor of being an Erdos
1 (published directly with Erdos). He
had already been a Straus 1 and an Einstein 2.

Joseph
Arkin had been appointed Senior Lecturer at West Point and for eight years had
collaborated with and mentored many faculty members in the Department of
Mathematical Sciences. He retired
from that position on 23 September 1994.

Joseph
Arkin has written over 50 articles which have appeared in numerous publications
such as the *Mathematics Magazine,
Fibonacci Quarterly, SIAM Review, Duke Mathematical Journal, Journal of
Recreational Mathematics, Notices of the American Mathematical Society, Canadian
Journal of Mathematics, Pacific Journal of Mathematics, and Mathematics and
Computer Education*. Among his
co-authors and collaborators are many distinguished mathematicians and
scientists including Paul Erdos, Ron Graham, E.G. Straus, Richard Pollack, Vern
Hoggatt, Paul Smith, V.E. Smith, Gerald Bergum, and Stephan Burr.
His co-authors at West Point include Bruce Porter, William Ebel, Charles
Kennedy, Edith Luchins, Lee Dewald, Rick Kolb, Frank Giordano, and Chris Arney.
His work is foundational in many areas and is, therefore, cited often in
works of other researchers. Joseph
Arkin has made over 50 presentations at professional meetings.
He has attended and presented papers at the American Mathematical Society
meetings, the meetings of the Metropolitan Section of the Mathematics
Association of America, International Conferences on the Fibonacci Numbers,
numerous number theory conferences, and Army Conferences on Applied Mathematics
and Computing. Joseph Arkin has
been a member of the New York Academy of Science, the Canadian Mathematics
Society, and The Calcutta Mathematics Society.
Some of Joseph Arkins, significant results are in the form of extensions
and generalizations of earlier classical works of great mathematicians.
Some of his papers of this type are: "An Extension of a Theorem of
Ramanujan," "A Note on a Theorem of Jacobi," "Researches on
Some Classical Problems," Exploded Myths," "New Observations on
Fermat's Last Theorem," "On Euler's Solution to a Problem of
Diophantus," and "An Extension of E. B. Straus' Perfect Latin 3-Cube
of Order 7." Most of his work was performed in several related areas of
number theory. In particular, Arkin
made tremendous contributions in the following areas: Fibonacci and other
recursive sequences, partitions, tilings, magic square and cubes, and Latin
squares and cubes. His Army awards
include the Certificate of Appreciation for Patriotic Civilian Service and the
Commander's Award for Public Service.

Today,
in this issue of *Mathematica Militaris*,
we give special tribute to the many contributions made by Joe Arkin and his
family. Joe lives at Ramapo Manor Nursing Home, Cragmere Road, Suffern, NY
10901. He's always happy to receive
greetings from fellow mathematicians, so drop him a line and express thanks to a
special person and a great mathematician.

This
note appeared in an issue of *Mathematics Militaris* dedicated to Joseph
Arkin: Founding Editor of *Mathematica Militaris* and an American Ramanujan. It is by David C.
Arney.