University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership May 2010 Perfect Numbers: Diana French University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons French, Diana, "Perfect Numbers:" (2010). MAT Exam Expository Papers. 13. https://digitalcommons.unl.edu/mathmidexppap/13 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Perfect Numbers: Why I’ve Decided I’m Glad I’m Only A Six, Because A Ten’s Not So Perfect After All! By Diana French Preface While this topic of “Perfect Numbers” was completely new to me (at least to the degree at which it is discussed within this paper,) I found it very intriguing and believe there is still much information and mathematical discovery in it for me. There were many historical points of interest, and I found it difficult to whittle them down to a manageable size for the intent of this paper. Likewise, there were many facts and peculiarities I found interesting and certainly worthy of consideration. However, to maintain anything close to a reasonable length of discussion as outlined in the guidelines for writing this paper, I found these things were best left out. A few of the ideas or facts that I found interesting are listed below. Also, I found I was left with some questions or ideas of my own that I simply did not have time at this point to explore. I’ve listed those as well as a reminder to myself of things I hope to return to at a later time. • There are only four perfect numbers between 1 and 3,000,000. • 652 is the only known non-perfect number whose number of divisors and sum of smaller divisors are perfect. (Divisors: 1 + 2 + 4 + 163 + 326 = 496 (a perfect number) and 1, 2, 4, 163, 326 & 652. There are “6” of them and 6 is perfect.) • Every even perfect number (except for 6) is the sum of consecutive odd cubes. For example, 28 = 13 + 33 = 13 + (2(3+1)/2 – 1)3 or 496 = 13 + 33 + 53 + 73 = 13 + 33 + 53 + (2(5+1)/2 - 1)3 • Theories and ideas concerning attributes of odd perfect numbers. (I find it quite interesting that so much can be known about something that may well not exist!) • Theorems and proofs regarding perfect numbers, the components of the formula used to find perfect numbers, etc. (Much of this was beyond my level of understanding or did not, in my opinion, add to the direction I chose to take this paper. One example, 2p – 1 is a Mersenne prime iff p is also prime.) • Modular arithmetic (I feel there is some connection to, or use for, modular arithmetic involving perfect numbers, the proof they cannot end in two or four, or something. I explored this to some degree, but simply ran out of time. Like many other parts of this research, it is something I hope to return to when time allows.) (NOTE: The title of this paper makes reference to a popular movie and expression from1979. The movie, “10” references a rating system for how perfect a woman (or man) is on a scale from 1 to 10, with 10 being considered perfect. Discovering the number 10 is not a perfect number but 6 is, I found I might have come upon a way to claim perfection at last!) The mathematical notation for this paper is somewhat tricky, as is often the case. As a means of clarifying the notation I’ve chosen, “MP” denotes a Mersenne prime number is used. The M is for “Mersenne”, the subscript p for “prime”. It simply means that 2 is raised to the power of a Mersenne prime when it is shown as a superscript (power) and it should be noted that formatting and font issues sometimes cause this notation to become somewhat misleading. As a general rule, the “MP” together simply means “Mersenne prime” regardless of the subscript p appearing to be a subscript of the term “2M” as it may seem. Hopefully, this will help avoid confusion between subscripts, superscripts and the use of “P” for perfect numbers and “p” for primes. 1 A perfect number can be defined as: A number equal to the sum of its proper divisors. With this in mind, it seems logical to consider the terms “divisor” and “proper divisor” and the role they play in determining perfect numbers. Divisor is a term (often called a “factor” of a number) that refers to a number that can be evenly divided into another number. If such a number exists, it is a divisor (or factor) of that number. (For example, 6 has factors of 1, 2, 3 & 6 since all these can evenly divide 6.) Proper divisors include all the divisors of a number (n) excluding that number (n) such that when you divide a number you will get a quotient greater than 1. (Thus, proper divisors of 6 are 1, 2 & 3 and NOT 6 because 6 ÷ 6 does not result in a quotient greater than 1.) Is 1 a proper divisor? My stance on this issue is based on the definition of “proper divisors”: The proper divisors of a number are all divisors excluding the number itself. By definition, 1 is included as a factor. This used to be considered in terms of “aliquot parts” of a number. (Aliquot parts of a number are proper divisors of the number that are smaller than the number. The aliquot parts of six are one, two, and three. While in definition this is exactly the same as “proper divisor”, looking at the word “aliquot” makes this a more reasonable explanation. The word aliquot joins the Latin ali (meaning "other") and quot (for “how many.”) Together they came to mean a part of something, in this case, a part of the number of which it is a factor. (“How many other parts are there?”) The "other" meaning of ali remains today in words like alias, alibi, and alien; very common “others”. The quot root remains in the word quotient. While this helps explain why one is a proper divisor, it also describes why the number itself is not a proper divisor: it is not an “other quotient”. For the quotient to be “other” (than the number itself) it would include anything that evenly divides the number “other than” that number. Now that we’ve defined perfect numbers (and cleared up the terms used in that definition) we can look at a few perfect numbers as a basis for what it really means for a number to be perfect. As 6 was used in the example above, let’s look again at its “proper divisors” and apply them to the definition of perfect numbers. Divisors of 6 remember are one, two, three and six. Proper divisors are one, two and three. Perfect numbers are numbers whose proper divisors have a sum equal to that number. Thus, since 6 = 1 + 2 + 3 (the proper divisors of 6) this number is perfect. There are 43 known perfect numbers. The first four perfect numbers are: 6 = 1 + 2 + 3 28 = 1 + 2 + 4 + 7 + 14 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 As you can see, the jump between perfect numbers is large and continues to get much larger as the list progresses. If you were to guess that the next (or 10th or 20th) perfect numbers would be very large numbers (increasingly so) you would be correct! This helps explain why there are relatively few known perfect numbers when you consider the largest known perfect number would require approximately 4,520 pages of this size to express all its digits! (Note: I actually tested this out! There are rd 18304103 digits in the 43 PN. 4050 digits will fit on a page with these margins. Thus, 18304103 ÷ 4050 = 4519.5316049382716049382716049383 pages.) 2 Having seen examples of perfect numbers, one might wonder where this idea of “perfect numbers” came from and who first discovered them. How are they found? Why are there so few of them and why are they so difficult to find? A brief history of “the perfect number search” will undoubtedly answer some of these questions. It is not known specifically who is to be credited with the discovery of the first 4 perfect numbers, but it is known that the ancient Greeks were aware of them. Having been “known” for well over 2000 years, knowledge and understanding of perfect numbers has progressed to the point today that people devote their time and careers to the quest for more knowledge and understanding: A great and vast search for perfect numbers. GIMPS (the Great Internet Mersenne Prime Search,) a computer networking project of over 4500 members, is one such endeavor that has been quite successful. The mystery and what was often perceived as “the magic” of the numbers known today to be perfect were first coined as “perfect” numbers by the Pythagoreans, or Pythagoras (569BC-475BC) and his disciples. However, the first known recorded mathematical result concerning perfect numbers occurred in 300 BC’s Elements written by Euclid. In Euclid’s Elements, he outlines a proposition surmising that a “double proportion” process resulted in perfect numbers.