These are actual student papers that were not designed to be web pages. They may contain historical, grammatical, mathematical, or formatting errors. These papers were graded using the criterion mentioned in the paper directions, and the writing checklist. The test review sheets and the WebCT tests are good indicators of the mathematics that was discussed in class during and/or after each presentation.
Beauregard Stubblefield

Beauregard Stubblefield



     “Success doesn’t come to you…you go to it (quotes, 1). ”  This was especially true of minorities in the 20th century.  Beauregard Stubblefield, an African American living in a time when blacks were not considered equal, rose above all racial barriers to achieve great success.   Beauregard’s determination to pursue a career in mathematics never subsided.  He would go on to overcome great obstacles, such as fighting for the right to be educated as well as working to pay his way through college.  He has, as a result of his hard work and dedication, become a great contributor to the world of mathematics.         

     Beauregard Stubblefield was born on July 31, 1923 in Navasota, Texas (transcript, 1).  Stubblefield, son of a watchmaker, was one of four children, two brothers Cedric and Elvin, and a sister Iris (transcript, 3).  As a child, Stubblefield’s father instilled in him the desire to pursue mathematics.  Every week, he would have Beauregard calculate the grocery bill, claiming each time that his calculations were correct.  His father would then tell him that because of this, he would “turn out to be a good mathematician (transcript, 2).” 

     Stubblefield’s devotion of achieving a higher mathematical status would not be easy.

During high school, Stubblefield was given a scholarship to Prairie View College, only to have it taken away and given to another student soon afterward (transcript, 5).  But Beauregard was determined to prevail.  After finishing an education at Booker T. Washington High School in 1940, Beauregard attended Prairie View College.  His teacher would be Dr. Clarence Francis Stevens, head of the math department.  Dr. Stevens soon recognized Stubblefield’s incredible mathematical talent.  Beauregard ended up making the highest score on the first exam even though he had not seen the Binomial Theorem or graphing in high school.  As a result, he was taken out of the class and would study one-on-one with Dr. Stephens (transcript, 2).

     While attending classes at Prairie View, Beauregard also worked using the watch making skills his father had taught him and his siblings, to pay for his education transcript, 4).  Stubblefield’s dedication would pay off, and in 1943 he received a bachelor’s degree and a master’s degree in 1945 from Prairie View University in Texas.  But looking for graduate work in Texas would prove to be unsuccessful.  Because the state of Texas did not offer him the graduate work he wanted, they would have to pay for his studies at the college of his choice (transcript, 2).  Stubblefield, as a result, sent in applications to four schools:  Columbia, Chicago, Michigan, and the University of Southern California.  After being told by the University in California that,  “they did not take anybody but their own students, ” and never receiving a reply from the other three schools, Beauregard took things into his own hands.  Because Stubblefield was determined to go to the University of Michigan, he decided to drive to the campus to find out why he did not receive an answer.  Surprised as to the boldness of this man, the University of Michigan felt as if they could not turn him down.  As a result, Beauregard Stubblefield became a student of Samuel Myers, a top mathematician at the university (transcript, 17).

     Soon afterward, the dean of students ruled that Beauregard was not a resident of Michigan because of the money he was receiving from Texas.  Stubblefield would have to choose whether or not he would keep receiving the money from Texas, and as a result pay more tuition (transcript, 24).  At the news of this, Stubblefield decided to take a small break from school and, once again, work as a watchmaker.  After working for six years at Hallis Jewelry store, Beauregard was able to save up enough money to continue his studies at the University of Michigan (transcript, 4).                                      

     It was then that he would meet his wife to be, Barbara Hill, a teacher in the Exeter public school system (transcript, 4).  Soon afterward they would marry and have a total of five children, three girls and two boys (transcript, 20). 

      In 1950, after his marriage to Barbara, Stubblefield would get back into his studies under the teachings of R.L. Wilder.  It was Wilder who would greatly influence Stubblefield and perhaps interest him on the subject of Topology, on which he would later receive a degree (1959) and write a book (transcript, 4). 

     Stubblefield continued to surprise people with his mathematical abilities, and in 1959, Stubblefield attended his first Society meeting under the recommendation of one of his professors.  He would speak on his solution to one of R.H. Bing’s problems, who was also in attendance at the meeting.  R.H. Bing at first did not believe that Stubblefield had actually solved his problem, but after he and Stubblefield sat down and talked about his solution, Bing realized, but did not admit that this talented young man had come up with a solution (transcript, 8). 

     While at Michigan, Stubblefield became very close to a lady from Liberia named Louise Austin (transcript, 12).  She would eventually help him to get a position as the head of the department of mathematics at the University of Liberia from 1952-1956.  This would be Stubblefield’s first real teaching experience (transcript, 13).  Fifteen years later, he would move to Boone, North Carolina to teach at Appalachian State University.  Bob Richardson, a professor at Appalachian, talks about discrimation towards blacks at this time and Beauregard Stubblefield whom he knew personally.  In 1966, Dr. Richardson helped in bringing the first black girl to Appalachian State University.  He talks about how when she went to look for housing, all she got told was that the apartments were already promised or filled.  “Stubblefield probably would have encountered the same thing, “ Richardson says.  The university knowing this, decided to put Beauregard in faculty housing (interview).  He would stay here five years, continuing his research on “number theory in search for lower bounds for odd perfect numbers (transcript, 14).” 

     Since this time, Beauregard Stubblefield has lived and traveled many places, but his life ambition has been to continue his work on number theory.  But the work he has already contributed will leave a profound impact on the world of mathematics as we know it.  He is attributed with several papers on mathematics including:  The Number of Topologies on a Set of Eight Elements, An Intuitive Approach to Elementary Geometry, Some Imbedding and Non-Imbedding Theorems for N-Manifolds, and Lower Bounds For Odd Perfect Numbers (lower, 295).     

            After receiving his M.S. in 1951, from the University of Michigan, Stubblefield became the Professor and Head of the Department of Mathematics at the University of Liberia at Monrovia from 1952-1956. From 1957-1959, he was a Research Mathematician at Detroit Arsenal. Upon receiving his Ph.D. in 1959, from the University of Michigan, he served as the Lecturer and National Science foundation Post-Doctoral Fellow at the University of Michigan in Ann Arbor until 1960. Stubblefield served as assistant Professor of Mathematics at Stevens Institute of Technology in Hoboken, New Jersey form 1960-1961. He was an associate Professor of Mathematics at Oakland University in Michigan from 1961-1967. From 1967-1970, he was on leave. When he returned, he was a Senior National Teaching Fellow at Prairie View, Texas until 1968. For the next year, he would be a Visiting Professor and Visiting Scholar at Texas Southern University. From 1969-1971, he was the Director of Mathematics in the Thirteen College Curriculum Program. Then in 1971, he served as a Professor of Mathematics at Appalachian State University in Boone, North Carolina until 1976. At the U.S. Department of Commerce in Boulder, Colorado, he served as Mathematician/EEO Manager from 1976-1981. In 1992, Stubblefield retired from the U.S. Department of Commerce with GERL/ERL/NOAA. Stubblefield, along with others, helped to establish NAM as an international organization. He was the person who, in 1978, approved NAM’s first major grant. In 1994, he received NAM’s Distinguished Service Award, and in 2000, Stubblefield became the eight recipient of NAM’s highest award, the Lifetime Achievement Award (mad).

Through his many years of research, Stubblefield had several publications on the subjects of geometry, trigonometry, and computer programming. The publications that he used for teaching were Informal Geometry, An Intuitive Approach to Elementary Geometry, and Structures of Number Systems. In 1962, he published his paper entitled Some imbedding and non-imbedding theorems for N-manifolds. Several years later in 1973, The number of topologies on a set of eight elements, was published. Finally in 1980, he published the paper Lower bounds for odd perfect numbers (beyond the googol) (mad).

            Though Stubblefield wrote and published numerous works, we are going to focus on his one paper entitled Lower bounds for odd perfect numbers (beyond the googol). The purpose of this paper is to “provide a newer method by which, when a natural number M is given, we can either (a) find an odd perfect number less than M or (b) determine that an odd perfect number less than M does not exist (lower, 211).”

            Natural numbers are the numbers that one would “naturally use for counting: 1,2,3,4… (natural).”  A positive integer n is said to be a perfect number if n is equal to the sum of all of its positive divisors, excluding n itself.  For example, let n = 6.  The divisors of 6, not including 6, are 1,2,3. Hence 1+2+3 = 6, and so 6 is a perfect number.  Now let n = 12.  The divisors of 12, not including 12, are 1,2,3,4,6.  Hence 1+2+3+4+6 = 16.  Since 16 does not equal 12, then 12 is not a perfect number. 

Prime numbers are those that have divisors of only 1 and itself.  This means that the sum of all its positive divisors, excluding itself is always going to equal 1 for all n that are prime.  For example, let n = 3.  Notice 3 is prime, so its only positive divisor, not including 3, is 1.  Since this is going to be the case for all prime numbers, prime numbers can never be perfect. 

Using the idea of perfect numbers, Stubblefield shows that if a square has a side that has a measure equal to an odd perfect number, then the area of that particular square is “beyond the googol.”  He claims in his paper that it is sufficient to let M = 1050 in order to show that the odd perfect number, that is the length of one of the sides of the square, is greater than 1050.  To show that this claim is adequate, let X = the length of one side of a square.  Hence, the area of the square is X2, which we will assume is greater than googol, or 10100 (lower, 211). 

area = X2 > 10100

odd perfect number = X = sqrtX2 > sqrt10100          

odd perfect number = X = X >1050

odd perfect number > 1050

Through the above calculations, we can see that Stubblefield’s claim is indeed correct. If the length of a side of a square is an odd perfect number than the length of the side is greater than 1050, which in turn means that the area of the square is “beyond the googol (lower, 211).”

Another idea that Stubblefield used to help him in his paper, was the fact that “Every positive integer n>1 can be expressed as a product of primes; this representation is unique, apart from the order in which the factors occur (Burton, 54).”  To show that this idea is indeed a fact, we will go through a proof of it.  n can be one of two things, a prime number or a composite number.  If n is a prime number, then there is nothing further to prove.  If n happens to be a composite number, then an integer d exists that satisfies d|n and 1<d<n.  Out of all the integers contained in d, choose p1 to be the smallest.  Hence p1 must be prime. If not, it would also have a divisor q with 1<q<p1, but then q|p1 and p1|n imply that q|n. This is a contradiction to the choice of p1 as the smallest positive divisor, not equal to 1, of n (Burton, 54).

Therefore, we may write n=p1n1, where p is prime and 1<n1<n.  We will have our representation if n1 is a prime number.  In the opposing case, the argument is repeated to come up with a second prime number p2 such that n1=p2n2. In other words,

n=p1p2n1,             1<n2<n1

Again, if n is a prime number, then we have nothing else to prove. Otherwise, we keep going by writing n2=p3n3, with p3 a prime:

n=p1p2p3n3           1<n2<n1

The decreasing sequence n>n1>n2>…>1 cannot continue forever, so that after a finite number of steps nk-1 is a prime pk. This leads to the prime factorization n=p1p2pk (Burton. 54).

            For the second part of our proof, the uniqueness of the prime factorization, we will let the integer n be represented as a product of primes in two ways:

n=p1p2pr=q1q2qs,          r<=s

with p and q being primes, and written in increasing magnitude so that p1<=p2<=…<=pr,q1<=q2<=…<=qs.  Since p1|q1q2qs, we know that p1=qk for some k, but then p1>=q1. Based on similar reasoning, we can say q1>=p1, hence p1=q1. When we cancel out the common factor, we get p2p3pr=q2q3qs.  We then repeat this process to get p2=q2 and this turns out to be p3p4pr=q3q4qs.  This process is continued. If the inequality r<s holds, we will arrive at 1=qr+1qr+2qs which is ridiculous, since each q1>1. Therefore r=s and p1=q1,p2=q2,…,pr=qr. This makes the two factorizations of n identical, as desired (Burton, 55).

            The name “perfect number” was given by the Pythagoreans, who often attributed mystical qualities to numbers.  For centuries, philosophers were not interested in the mathematical properties of perfect numbers, but were more concerned with their mystical or religious significance.  Saint Augustine used the idea of perfect numbers when talking about the creation of the world.  He said that God chose to create the world in 6 days instead of 1, because the perfection of his work is symbolized by the perfect number 6.  Another argument of the perfection of the universe was by the early critics of the Old Testament.  They claimed that the number 28, also perfect, represents the perfection of the universe since 28 is the number of days it takes the moon to circle the earth.  Alcuin of York, an 8th century theologian, observed that the whole human race came from the 8 souls on Noah’s Ark and that this second Creation is not as perfect as the first, 8 being an imperfect number.  The ancient Greeks only knew of 4 perfect numbers. P1 = 6, P2 = 28, P3 = 496, P4 = 8128. The Greeks claimed that these numbers were formed in an “orderly” fashion, one among the ones, one among the tens, one among hundreds, and one among the thousands.  Based on this small amount of evidence, two things were inferred:

  1. the nth perfect number Pn contains exactly n digits
  2. the even perfect numbers end, alternately, in 6 and 8.

Both of these claims were wrong.  A perfect number with five digits does not exist. The next perfect number is P5 = 33,550,336.  While P5 does end in a 6, the next perfect number P6 = 8,589,869,056 also ends in a six, not an 8 as thought by the Greeks (Burton, 250-254).


Since almost the beginning of mathematical time, there has been a problem of determining the general form of all perfect numbers. Thanks to Euclid and Euler, we now know the format for even perfect numbers. If 2k-1 is prime (k>1), then n = 2k-1 (2k-1) is perfect and every even perfect number is of this form. Now that the form of even perfect numbers is known, the problem of finding even perfect numbers is reduced to the search for prime numbers in the form 2k-1.  One of the difficulties in finding additional perfect numbers is the unavailability of tables of primes.  This raises the question of whether there are infinitely many prime numbers that are of the form 2P-1, with p being prime (Burton, 251-254).

Looking at how large the sixth perfect number already is, one can begin to see just how rare perfect numbers really are.  Though we do know they are rare, it is not yet known whether there are infinitely many, or finitely many perfect numbers.  This of course would depend on the answer to the above question of whether there are infinitely many prime numbers that are of the form 2P-1, with p being prime (Burton, 252-254).  Though perfect numbers are very rare, number theorists today find them useful for things such as string theory (string). 

The question of the existence of odd perfect numbers has yet to be answered. For years people have been trying to answer this question. There has not been a single odd perfect number discovered yet, but there is also not any proof saying that they do not exist. We are expected to have an answer to the question of their existence sometime in the near future.

            Stubblefield is one of the many mathematicians that have contributed in trying to determine if odd perfect numbers exist.  In order to try and prove or disprove the existence of odd perfect numbers, mathematicians have resulted to the use lower bounds.  The idea behind this approach is to find a number n and prove that there does not exist any odd perfect numbers that are less than n.  Hence, n being the lower bound.  In Stubblefield’s paper, he chose n=1050. He goes on to prove that indeed there are no odd perfect numbers that are less that 1050.  As a result, he had found a new lower bound for odd perfect numbers, 1050 (lower, 211-222).        






















     Burton, David M. Elementary Number Theory, Second Edition. Wm.C. Brown Publishers. Dubuque, Iowa (1989), 55-56,250-254. (Burton) Very useful for history of perfect numbers, definitions and proofs.


     Famous Quotes By Black Americans.

(quotes) We used this to get a good quote that would apply to our writer.


     Interview with Bob Richardson, professor at Appalachian State University.

(interview) This was very useful in getting information about Beauregard when he worked here at Appalachian State University.    


     Lower bounds for odd perfect numbers (beyond the googol), Black Mathematicians and Their Works, Dorrance & Co. Ardmore, Pa. (1980), 211-222. (lower)  Was useful for getting ideas that Stubblefield studied.


     Mathematics of the African Diaspora: Beaurguard Stubblefield. (mad)

Very useful for publications and places of employment.



     Natural Numbers and Whole Numbers.  (natural) Used for the definition of natural numbers.


     String theory and quantum cosmology. (string) Used to determine what perfect numbers are used for today.



     Transcript of an Interview with Stubblefield.  Conducted by Dr. Albert Lewis.  Educational Advancement Foundation.  Austin, Texas(1999), 1-26. (transcript)

Was very useful in learning about the life of Beauregard Stubblefield.