**The
Quaternion**

*The Newsletter of the Department of Mathematics,
USF-Tampa*

Volume 20: Number 1 Fall, 2005

**Transitions**

**Ken Pothoven** has
retired after 35 years at USF.

He got his bachelor’s degree from

Although he remained active in research, in real analysis and differential equations, he is best known for his service to mathematics education. He was very active in the MAA, helping organize Suncoast meetings at USF since 1978, and helped with a vast number of community programs and educational initiatives, ultimately becoming Secretary-Treasurer of the Florida MAA Section from 1999 to 2003. Meanwhile, closer to home, he took over the Center for Mathematical Services in 1994 (from which he recently retired) and was very active in the calculus reform movement.

We are grateful for his contributions, and we wish him well on his further adventures.

If
Size Matters, then how Large are the Primes?

*by Boris
Shekhtman*

Counting primes is
not like counting ballots in an election: if we wanted to know what proportion
(or “density”) of the integers are prime, we learn little by counting them. There are countably many primes and
countably many non-primes. To find
the density, one must do what the exit pollsters attempt: measure the density
itself. We do this by estimating
how many of the first *n* integers are
prime, and let *n* go to
infinity.

Applying this sampling
logic to the size of the primes gives the “The Prime Number Theorem”: in 1896,
Hadamard and de la Valle Poussin in 1896
independently verified a formula for the asymptotic density of primes that are
less than some given number:

(#primes < *n*)/*n* ≈ 1/(log *n*).

But
then, the density of Primes, lim1/(log *n*), goes to zero. To bad for primes, but
on the positive side, this is as large a zero as there is.

** **In nature, small
animals are, relatively speaking, faster then the giants. One could consider
size as a quantity inversely proportional to speed. The slower a sequence speeds
to infinity, the slower the reciprocals tend to zero, and the better the chances
that the sum of the reciprocals
diverges.

The sum of reciprocals of primes is infinite: the primes tend to infinity slow enough
to be LARGE.

So the density of primes is both zero and large. We need a tiebreaker.

**
**In 1927, B. L. van der Waerden
published his famous theorem: In any finite partition of integers, one of the
sets of the partition contains arithmetic sequence of arbitrary large finite
length. He thought of a large set as a set that contains arithmetic sequences of
arbitrary length.

An arithmetic sequence
(of length *n*), is of the form *a*, *a* + *d*, *a* + 2*d*, *a* + 3*d*, …,*a* + *nd*. For instance, the set of natural
numbers is an infinite arithmetic sequence with *a* = *d* = 1. Delete every millionth number and
you have a set, almost as large as naturals, but not an arithmetic sequence.
Nevertheless, it contains arithmetic sequences of arbitrary length. To further
elaborate on the relationship between size and arithmetic sequences, in 1936,
Erdös and Turan conjectured that every set with positive (“upper”) density
contains arithmetic sequences of arbitrary length. E. Szemeredi proved the
conjecture in 1975. So, the sets that are large in terms of density are large in
terms of arithmetic. Where does it leave the Primes? In limbo for thirty years.
Well, 3,5,7 is a sequence of three primes of constant difference two. The primes 5, 11, 17, 23, 29 form an arithmetic sequence
with constant difference 6. The world record is an arithmetic sequence of
TEN primes discovered by Manfred Toplic in 1998: Start with the prime 100,996,972,
469,714,247,637,786,655,587,969,840,329,509,

324,689,190,041,803,603,417,758,904,341,703,

348,882,159,067,229,719, and
use a constant difference of 210. Finally, last year, Ben Green and Terence Tao
used the Semeredi’s theorem in combination with a “transference principle” and
48 pages of technical mathematics to establish that the primes *do* contain
arbitrarily long arithmetic progressions. Joy to the world, the Primes are BIG!
This macho melodrama is not over.

Erdös conjectured that if a set of integers has an infinite sum of its
reciprocals, then the set contains arithmetic sequences of arbitrary length. Try
it. Maybe you will get lucky.

**The Nagle
Lecture: Andrew Odlyzko on
Cybersecurity**

Andrew Odlyzko gave the 13th R.
Kent Nagle Lecture on *Cybersecurity,
Mathematics, and the Limits of Technology* to an audience of about 170 people
on February 24. Professor Odlyzko,
Director of the

Humans are good at
coming up with cumbersome security systems that humans then finesse in order to
get things done. Secretaries
routinely forge and fax signatures, while lawyers write laws and contracts with
deliberately ambiguous wording to preserve slack. The point is that humans are supposed to
use their “common sense.”
Odlyzko’s example is of someone who asks a neighbor to “let the plumber
in to fix the leaky faucet”: a sensible neighbor would be presumed to know that
something was wrong if the plumber started removing
furniture.

Until recently,
security problems were the usual embezzlement, bad checks, hold-ups, etc. Even now, most scams rely on tricking
users to reveal credit card numbers for phony security checks, or sending money
to cover handling costs for Nigerian lotteries. There are some technical security
problems, such as the “buffer overruns” that have facilitated most virus and
worm invasions in the last three decades; yet it was humans that have known
about this problem and done little about it.

Odlyzko’s point is
that the problem is *us* goes beyond
security. For example, poorly
written software may be an irritant for users, but the necessity for endless
upgrades provides job security for the code writers. And so, as long as humans are involved,
there will be security issues...

**Center /
Mathematical Services**

The Center continues to be
involved in outreach and service activities to the Suncoast
Region.

In the summer of 2005,
the Center conducted two programs for gifted and high-ability secondary
students. Both programs ran concurrently from June 6 through July 8, from 9 a.m.
to 3 p.m. weekdays. This was the 27th year that the Center has conducted such
programs. The programs had a total of 32 students who were taught courses in
mathematics, computer science, and environmental science. All students were
taught these courses in formal class settings, but they also participated in
laboratory exercises in computer science. A grant from the

**The Math
Club**

The Math Club – the joint USF
Student Chapter of the Mathematical Association of America (MAA) and the Florida
Epsilon Chapter of Pi Mu Epsilon (PME) – met fourteen times during the academic
year. The meetings are mostly attended by math undergraduate students. There was
free pizza and sodas and a speaker at every meeting. The talks were given by
both faculty and undergraduate and graduate students and topics ranged from
applications of mathematics to biology and chemistry to proofs in mathematics,
interesting geometry problems and math games among others.

In additional
news:

PME again hosted the
Spring and Fall Hillsborough County Math Bowls with all 23 county high schools
sending team and individual student competitors; top honors went to **H. B. Plant High School** winning top
honors in the overall competition.

At the MAA Suncoast
Regional Meeting in December 2004 at **Matt Williamson**, one of four USF math
undergraduate students attending, delivered the student presentation *Inversion and Geometry: An Interesting
Technique Not Usually Taught in Geometry Class*. And at the 2005 Joint meetings of the
MAA and the FTYCMA in February 2005 at

Two USF undergraduate students attended
the 2005 Joint Meetings of the MAA and the AMS in January 2005 in

At the 2005 St. Pete
College Mathematics Awareness Conference, undergraduate students **Keith Grizzell** and **Nicole Hooper** won a prize for solving a
math problem posed by one of the speakers at the
conference.

Our
PME chapter inducted eleven new members this year. The induction ceremony was attended by several inductees’ parents and relatives
as well as mathematics faculty members. The keynote speaker was Dr. Gordon Fox
of the biology department.

The 2005 PME
Outstanding Scholar was **Anand Bhat**.
After finishing his math major last December, he has already started graduate
studies in our department. Upon being named for this award, he delivered a
well-received math club talk titled “Magic Squares- Some Math and Some Magic,”
and he received a plaque at the PME banquet.

**Darshit J. Patel** won the 2005 USF
Council of Honor Societies Academic Achievement Award for the second year in a
row. As president of the PME
chapter, he represented the chapter at the joint MAA Math / PME national meeting
in August.

**Student
News**

Four students were awarded
doctorates during the 2004-’05 academic year: Djiby Fall (under Y. You; *Longtime Dynamics of Hyperbolic Evolutionary
Equations in Unbounded Domains and Lattice Systems*), David Edwin Kephart
(under N. Jonoska; *Topology, Morphism,
and Randomness on the Space of Formal Languages*), Ferenc Tookos (under V.
Totik; *Hölder Continuity of Green’s
Functions*), and Norbert Noupeyou Youmbi (under A. Mukherjea; *Contributions to Harmonic Analysis and
Probability Theory on Semihypergroups*).

Thirteen students were
awarded Master degrees: Angela
Angeleska, Jodi Louise Barlow, Lisa Marie Stephenson Borzewski, David Jeffrey
Bueller, Stacey Lynn Cummins, Rajesh Ganesan, George W. Kimber, Jr., Gayathri
Mahalingam, Meagan Nicole McNamee, Robert David Mitchell, Ena Lynette Salter,
Janet Hester Samuels, and Anupama Tippabhotla.

Twenty-five received
Bachelor’s degrees: Hashim Ahmed
(Cum Laude),

Richard Arriaga, Anand Bhat
(Magna Cum Laude), Scotty Boutte, Judi Charley-Sale, Natalie Davis (Magna Cum
Laude), Jennifer Ezell, Justin Feller, Joshua Felton, Brian Frasier, Fred Gore
(Magna Cum Laude), Keith Grizzell, Alex Guevara (Cum Laude), Jessica Halsell,
Princess Harris, Christopher Hollander, Tanya Jones

Thomas Joyce, Avni Kardani,
Jason Karol, John Knisley, E'Leon Mills, Robert Rienzi, Gregory Thole (Cum
Laude), and Daryl Williams.

**We’d Like to Hear from YOU!**

The Department of Mathematics would like to hear from alumni, friends, collaborators, members of the community, and fellow explorers of and guides to the world of mathematics.

Contact us at: 974-2643, or
fax 974-2700. E-mail
<mathdept@math.usf.edu>. We
have a web-page at <http://www.math.usf.edu/>. Snail-mail address is Department of
Mathematics,

**Correction**

The
appeal entitled *The Continuing Crisis*
in the print version of the 2005 Quaternion was actually recycled from previous
issues, and out of date. We are pleased that since then, we have been able
to greatly reduce our dependence on adjuncts in lower division courses, and that
the Nagle Lecture Series has been reactivated. We apologize for
the outdated item, and we hope we did not alarm
anyone.