Thomas P. Dence, Ashland University (


Paul Bruckman (left) with Thomas P. Dence

Many readers of The Pi Mu Epsilon Journal may be unfamiliar with the name Paul Bruckman, yet there will be a significant set of readers who know the name well.  Starting around 1993, Paul has been solving (and also submitting) problems posed in The Problem Department of the Journal.  This, by itself, is not as remarkable as the fact that Paul has solved virtually every problem during this time period! Paul, though, will argue that this is an overstatement, and that there have been many problems that he didn’t solve. His record, though, parallels taking a 16- year exam of roughly 300 problems (not too many easy ones) and scoring a grade of A+.  Furthermore, if you think this streak is amazing, and rightly you should, consider the additional fact that Paul has solved practically every problem from The Fibonacci Quarterly since 1976!  One, in fact, can do a YAHOO search for The Fibonacci Quarterly, and clink on the link for Problem Solutions, and find approximately 737 problems (over 1000 is more accurate), proposed by other readers, that Paul Bruckman has solved. There is also a significant number of problems that Paul had proposed for the readers of The Quarterly, and subsequently solved himself.  It makes sense, therefore, to place Paul among (if not right near the top of) the country’s all-time great problem solvers.  

This author recently traveled to Vancouver, British Columbia, to meet with Paul.  He had moved a short while ago from the west coast of California to Washington and then to British Columbia in order to be close to the water’s edge, something that his wife Lynn was in full agreement.  Settling in the small town of Sointula, on a small island off the northern tip of Vancouver Island, Paul was content to work on his mathematics with minor interruptions, until the Spring of 2009 when health problems forced him to relocate to the big city of Vancouver, where ample medical help was available.  I then met him and his wife during early August at the townhouse (belonging to a friend) that they were renting.


TD:      First, tell us a little about your early life.
PB:      I was born in 1939 in Florence, Italy, where I lived for 9 years.  Clearly this was prime time for World War II, and I vividly remember some local destruction, such as some occasional bombing and the Nazis blowing up nearly every bridge in town.  On the bright side, though, I was exposed to many fine things about Italian life.  Cooking, for example, is one of my favorite endeavors as I can make from scratch a multitude of pasta dishes.  I became enthralled with the fine arts, such as music and literature.  My grandfather was a master violin maker (as well as an artist of some renown), and I would spend countless hours watching him cut the wood, piece it together, stain and varnish it, and pluck away on the strings to get the right sounds.  I’m also a lover of classical music, and I’ve even had one of my own poems, “Constantly Mean,” published, of which portions of it are quoted in Mario Livio’s engaging book “The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number.”  The poem is not about music, but is a mathematical poem, as are two other poems (one about pi, and one about e) that I’ve written.

Around 1947 my parents and I emigrated to the United States.  My mother came over first, then Dad and I followed.  Unfortunately my parents were splitting up, so Dad stopped in New York to relocate and I continued to Cincinnati to meet my Mom.  She and I, and my new stepfather, headed to the north Chicago area, where Highwood, Illinois, was to be home for about 21 years.

TD:      Take us through your educational background.
PB:      As it turned out, I was a pretty good student back in Florence, where I advanced faster than most students.  Likewise, in Highwood, I scored well during my high school days.  The mathematics classes I took consisted of geometry, trigonometry, algebra, and spherical trigonometry.

TD:      That’s interesting because that was the same curriculum I took in high school in Toledo, Ohio.  I’m afraid spherical trigonometry has disappeared from today’s curriculum, and plane trigonometry has been demoted from a semester long course to a chapter or two out of a precalculus text.  The geometry has also lost it’s proof-oriented and deductive reasoning based regimen, much to Euclid’s chagrin.
PB:      Mathematics, interesting enough, wasn’t my main love in high school – that spot was occupied by Biology.  I was fascinated, and still am today, by animals and insects.
Creatures like millipedes, cicadas, tomato worms, butterflies, and tarantulas amaze me by their beauty and how they function.  I recall marveling as the lovely lightning bugs, which are native to the Midwest, light up the summer evening sky with their on-again, off-again light switches.

As I began my college career in 1957, academic life became a little more discon-tinuous.  I started at the University of Chicago where I spent just one year.  But it was here, with my first exposure to the calculus, that my mathematical life really started to blossom.  Calculus had a way to give such precise rationale as to why certain things worked, especially, as I recall, with the conic curves.  The following two years were spent at Illinois State Normal, and then one year at the University of Illinois at Champaign-Urbana.

About this time I got married and needed to earn income, so I left school without a degree, and started employment with an actuarial consulting company working mostly on pension plans.  This, in fact, was going to be my line of work until I retired in 2001.

TD:      You were soon to be drafted into the army?
PB:      Yes, that happened in 1963.  Within the next year I was stationed over in Germany and, as luck would have it, I’m refreshing myself one evening at one of Germany’s finest watering establishments when I meet this lovely young Canadian gal who was hitchhiking/working her way around Europe for the summer with a girlfriend (whose Vancouver townhouse we are staying in right now).  Two years later we are both back in the states, and we got married, and Lynn has been with me for all 44 years since then.  We have raised 5 children, 4 daughters and one son.  I have two children by my first marriage.  After marriage, I then resumed my actuarial work in Illinois, until 1968, when we then decided to move to San Francisco.  Lynn and I have this corny little joke between us that to move to California was predicated on the fact that I couldn’t get a
good enough haircut in Illinois.  I tried cutting my hair, and she would cut it, and my head would wind up looking like it belonged on a concentration camp refugee.  I couldn’t go to work looking that that, so we decided to move westward. 

TD:      How long were you in California?
PB:      I stayed there four years, and then decided I wanted to return to Illinois, so I spent 1972-76 at the University of Illinois at Chicago where I earned both a B.S. and M.S. degree.  I was just a whisker away from a Ph.D. degree, but I decided to forgo it (much to the dismay of Lynn) and return to the lucrative work force, which would eventually relocate us back to California. 


TD:      Wasn’t this the period of time, though, when you became interested in the Fibonacci numbers?
PB:      Yes, my first paper published in The Fibonacci Quarterly was in 1972, followed by four more within the next three years, and I began to solve problems from the Problem Department in 1975.  Actually I only solved about half a dozen problems that year, but then in 1976 I started rolling and was solving practically every one.

TD:      How did you really come to get stated with The Quarterly?  Was there someone who influenced your work by pointing you in this direction?
PB:      It’s been so long ago that it is hard to tell.  I think perhaps a colleague at work, or a friend, suggested I should look at the journal.  But once I got started, I was most definitely helped by Verner Hoggatt, Jr., who was a professor (I think) at the University of Santa Clara.  He literally took me under his wing and encouraged me in many ways to pursue further work.  He would correspond by mail with me many times and just pepper me with problems and ideas to consider.  Verner was really an inspiration for me, and I am most grateful for his encouragements and persistence with my mathematical success.

TD:      What about the PME Journal?
PB:      That was just a case of my looking around for some additional problems to work on, and a lot of the problems in this journal are really very creative and interesting – so I enjoy working on them.  I have at times solved problems from The American Mathematical Monthly and the Missouri Journal of Mathematical Sciences, but not really too many.

TD:      Are there certain kinds of problems, be it limit problems, evaluating integrals, or proofs by induction, that when you first see them you lick you lips and say “Goody, I
know how to do this.”? Are there also kinds of problems that are especially tough, or displeasing, to you?
PB:      Well, I do enjoy the type of problem where I have to do some calculations, cause I like to search for patterns.  So if I’m trying to sum a series, I might first try and get a
handle on an expression for the n-th partial sum.  I really do have a wide interest in mathematics, although I’m not particularly crazy about problems of logic, or things that involve formalistic expressions.  Probability problems are typically tougher and can be tricky at times, but I still enjoy them.

TD:      Do you have a stockpile of identities and/or inequalities that you refer to when needed?
PB:      No, not really.  I might refer to a book of tables or formulas if I have to, otherwise I would just try and reproduce and figure out, or derive from scratch, what I needed.  But then my solution only gets recognized, not published in its entirety, in the journal cause it is too long.

TD:      Do you like it when your solution does get published?
PB:      Yes, I really do!

TD:      Then you should make your solution shorter!
PB:      Your point is well taken.

TD:      How often does your first line of attack on a problem work, and are there times when you have to try two, three, or four methods of attack?  This should be a most valuable lesson for students.
PB:      By all means I have to try multiple lines of attack quite frequently.  You need to know, for instance, different ways to establish that a sequence is increasing.  If I were to give some advice on “problem solving techniques” to students today, I would say

  1. learn as many different mathematical strategies as possible,
  2. have broad knowledge in many branches of mathematics, and
  3. most importantly, have patience and persistence!

There are going to be so many times when your line of attack doesn’t become fruitful, but DON’T GIVE UP!!  Keep trying.  You’ll be surprised how often something beneficial will eventually develop.  I find that I fail often, but I go after it like a bulldog, especially if I like the problem.

TD:      Sometimes when people work long enough on a problem their mind continues to work on it even when they are doing something else, like sleeping or watching television.  Has this ever happened to you?
PB:      Oh yes, it sure has.  Several times it has occurred when I’ve been in the bathroom!

TD:      So you mean sometimes you do your best work when sitting on the throne?
PB:      Yes, sure, it’s a real quiet place, and I get to relax and take my time.  Several times solutions have come to me while I’ve been sleeping and dreaming. 

TD:      Just out of curiosity, have you ever been asked to help coach the U. S. Olympiad team?
PB:      No.

TD:      Have you ever been asked to speak to any special group of students, like maybe a student math club or a student math chapter?
PB:      No.  I’ve never been asked, but I might entertain the idea if the circumstances were right.  I don’t have a lot of teaching experience, nor a lot of practice with oral communication, but to be confronted by a group of eager-beaver students could be real energetic.  I would be honored to have the request.

TD:      Do you notice any special difference in the type of problems in the two journals.
PB:      Outside of the obvious distinction that the majority of problems in The Quarterly
are Fibonacci related, I do notice that there are a fair number of geometry problems in the
PME Journal, and that’s nice; also more problems on probability.

TD:      There seemed to have been a special set of people who, early on, were highly frequent problem solvers in The Quarterly.  Names like George Berzenyi, Verner Hoggatt Jr., Bob Prielipp, Sahib Singh, Charles Trigg, Herta Freitag, Leonard Carlitz, David Zeitlin, and Gerald Bergum were some of these avid problem solvers.  Did you ever get to know any of these people on a more friendly basis?  Did you view them as “friendly rivals?”
PB:      I didn’t really know any of them except Verner Hoggatt, as I’ve already mentioned.  He and I collaborated on a couple of papers, and that’s how I acquired an Erdös number of two.  I did have a little tiff with Bergum once when he was the editor of The Quarterly.  Seems like he didn’t approve of the argument in one of my papers.  But no, I didn’t view any of these people as rivals.  There are some names that you could have mentioned that are more reflective of recent efforts.   Now, though, I do think that one of the current frequent problem solvers for The Quarterly is trying hard to “outdo me.”  He mentioned something once about creating a little bit of rivalry.

TD:      Are you by chance familiar with The Pentagon, which is the official journal of Kappa Mu Epsilon, the other mathematics honorary society in U. S. colleges and universities?  This, in fact, was the journal that was handed to me when I was a freshman at Bowling Green State University by my calculus professor, J. Frederick Leetch, who suggested I try solving some of the problems in it, and hence my interest in solving
problems was turned on.  As a side note, the Problem Section editor in those day was F. Max Stein, a professor at Colorado State University, and he wound up giving me a graduate assistantship when I applied at CSU for entry into their Ph.D. program in 1971.
PB:      No, I’m not, but now that you mention it I’ll certainly look into it.

TD:      There is an interesting story about an incident of yours with The Fibonacci
Quarterly – you probably know about it.  In 1993, roughly 20 years after you had been solving problems in The Quarterly, a problem, B735, was sent in to the editor by Curtis Cooper and Robert Kennedy, requiring readers to show that a particular linear recursive sequence of order 15 generated nothing but perfect squares.  The editor, believing you would finally be stumped on a problem, proclaimed “Ah ha, we finally have got him.”  But, within a week after the journal hit the newsstands, your solution appeared on the editor’s desk.  Sounds like a page out of the book on Newton!  Tell us about this.  What was your approach in solving it?
PB:      I guess I heard a little about this, maybe I read something on it.  It was kind of a cute story.  Unfortunately, I don’t remember how I solved it.  Probably I looked for a pattern.  I could look it up in my files, but all my mathematical papers are in boxes over in Nanaimo, on Vancouver Island, since we are in the process of moving soon into our new residence over there.

TD:      Has “problem solving” been instrumental in your doing mathematical research?
PB:      Absolutely, no question.  There’s a positive correlation between solving problems and doing research.  Both involve lots of the same skills and analytic tools.  In order to learn mathematics you have to do a lot of problems.  Also, frequently, the idea for a research project is the outgrowth of having worked on a similar problem.  I do have over three dozen publications, most of these in The Fibonacci Quarterly and The Pi Mu Epsilon Journal, and several of them involve generalizations of problems that I had worked on.


TD:      Tell us about some of your current research.  Seems like I might have heard that you had been working on the Twin Primes Conjecture, or the Syracuse problem.  Oh oh, I see a big smile on your face!
PB:      I’ve worked on the Twin Primes, but I haven’t been able to solve it.  It’s proving rather difficult so far.  But what would you say if I told you I’ve solved the Collatz Conjecture (this is also called the Syracuse Problem or the 3n+1-Problem by various authors)?

TD:      I would say WOW!  Really!  Have you really solved it?
PB:      I sure think so, unless there is a flaw in my argument that I’m unaware of.  The paper is currently under review by Annals of Mathematics.  And then what would you say if I told you that I’ve solved the Riemann Hypothesis?

TD:      I would say that I have some lakefront property in Nebraska to sell you!  Are you kidding?
PB:      No, I’m not.  This paper has also been submitted to the Annals, and I’m waiting to hear from them.  They have actually had it for quite some time now, and the longer I go without hearing from them the more encouraged I am that my proof is correct.  So we’ll see what happens.  I happen also to be close to proving Goldbach’s Conjecture, where every even number greater than two can be written as the sum of two primes, but I don’t quite have it finished yet.

TD:      Wow, I’m pretty speechless.  I wish you the best of luck.  I offer you  my utmost congratulations – I am supremely impressed. When you become famous I hope you remember who are your friends! In closing, I want to thank you for taking the time to allow me to visit you.  It couldn’t have been easy, what with your current medical problems.  I would say Lynn (who has years of nursing experience) has been taking good care of you.  Now, I didn’t fly all the way here from Ohio empty-handed.  I came equipped with a couple of problems for you to work on.  One is mine that deals with the transcendence of a series of reciprocals of certain termsin a Fibonacci related sequence, and the other problem, courtesy of my brother Joe, seeks a series representation for the first generalized Euler constant.  I’ll show these to you while we’re having the ice cream, strawberries, and Canadian blueberries that Lynn is getting ready to serve us.
PB:      I thank you for coming and next time you return, plan on staying longer.