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The Schnirelmann Density of the Set of Deficient Numbers

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The Schnirelmann Density of the Set of Deficient Numbers THE SCHNIRELMANN DENSITY OF THE SET OF DEFICIENT NUMBERS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Peter Gerralld Banda 2015 SIGNATURE PAGE THESIS: THE SCHNIRELMANN DENSITY OF THE SET OF DEFICIENT NUMBERS AUTHOR: Peter Gerralld Banda DATE SUBMITTED: Summer 2015 Mathematics and Statistics Department Dr. Mitsuo Kobayashi Thesis Committee Chair Mathematics & Statistics Dr. Amber Rosin Mathematics & Statistics Dr. John Rock Mathematics & Statistics ii ACKNOWLEDGMENTS I would like to take a moment to express my sincerest appreciation of my fianc´ee’s never ceasing support without which I would not be here. Over the years our rela­ tionship has proven invaluable and I am sure that there will be many more fruitful years to come. I would like to thank my friends/co-workers/peers who shared the long nights, tears and triumphs that brought my mathematical understanding to what it is today. Without this, graduate school would have been lonely and I prob­ ably would have not pushed on. I would like to thank my teachers and mentors. Their commitment to teaching and passion for mathematics provided me with the incentive to work and succeed in this educational endeavour. Last but not least, I would like to thank my advisor and mentor, Dr. Mitsuo Kobayashi. Thanks to his patience and guidance, I made it this far with my sanity mostly intact. With his expertise in mathematics and programming, he was able to correct the direction of my efforts no matter how far they strayed. His never wavering confidence in me drove me through many a pot of coffee and finally paid off with this thesis. iii ABSTRACT The goal of this discussion is to determine the Schnirelmann density of the deficient numbers, which is not currently known. The Schnirelmann density for deficient numbers is defined as σpDq “ infrDpnq{ns, where Dpnq denotes the cardinality of the set of deficient numbers not exceeding n, and inf is the infimum operator. This density is of particular interest because of the useful properties that it possesses. The Schnirelmann density of the deficient numbers can be found by finding a local minimum density in an initial interval of the natural numbers and then lower- bounding the density over the rest of the set of natural numbers. The lower bound for the density of deficient numbers is equivalent to an upper bound for the density of the complement set of the deficient numbers, the non-deficient numbers. To achieve our goal we incorporate Felix Behrend’s procedure for bounding the density of the non-deficient numbers with some improvements to reduce the effects of the approximations he made. With this information about the two intervals, it is expected to arrive at the Schnirelmann density of deficient numbers. iv Contents Signature Page ii Acknowledgements iii Abstract iv List of Tables vii List of Figures viii 1 Exposition 1 1.1 Where did it all start? ......................... 1 1.2 Density in Mathematics ........................ 3 1.3 The Schnirelmann Density ....................... 5 1.4 More from the Number Theorist’s Handbook . 6 2 Developing The Bound 11 2.1 The Candidate Number ........................ 11 2.2 Bounding the Contributors ....................... 12 2.3 Calculating the Density of Non-Deficients . 17 2.4 Revisiting the Bound .......................... 19 v 3 Results and Interpretations 22 3.1 Initial Bounds: Behrend’s Result ................... 22 3.2 An Illustration of the Bounding Method . 24 3.3 The Refining Process .......................... 25 4 The Schnirelmann Density of Deficient Numbers 31 4.1 Improvements on Bound and Execution Time . 32 4.2 Second-Order Terms and Further Research . 35 Bibiliography 38 Appendices 39 A Deficient Data 39 B Programs 41 B.1 Program 1 ................................ 41 B.2 Program 2 ................................ 44 vi List of Tables 3.1 Conditions for improvement ...................... 26 3.2 Revised conditions for improvement . 28 3.3 Final conditions for improvement ................... 28 4.1 τ Specific Densities of Non-deficient Numbers . 31 4.2 Improved τ Specific Densities of Non-deficient Numbers . 34 4.3 Further Improved τ Specific Densities of Non-deficient Numbers . 34 A.1 Density of Deficient Numbers up to n ................. 39 A.2 Density of Deficient Numbers up to n continued ........... 40 vii List of Figures 2.1 Density of Deficient Numbers ..................... 11 3.1 Partition Scheme ............................ 24 3.2 Cardinality Partition Scheme up to n ................. 25 viii Chapter 1 Exposition 1.1 Where did it all start? Euclid (300 B.C.) made mention of different types of numbers in his thirteen- book treatise called The Elements. This collection of mathematics is considered by many to be one of the most influential works in the history of mathematics. In the Elements, Euclid realized the principles of what is now called Euclidean geometry from a small set of axioms. This collection contains works on perspective, conic sections, spherical geometry, number theory and rigor. Euclid talks about a specific type of number called perfect or complete. Nicomachus (100 C.E.) continued this discussion and categorized the natural numbers into three types.[1] These numbers have the property that the sum of their proper divisors either is greater than the number itself, is the number itself, or is less than number itself. The perfect numbers are the numbers where the sum of their proper divisors is the number itself. To understand mathematically and symbolically what a perfect number is, consider the following definition and developed notation. 1 (Note that N denotes the number theorist’s set of natural numbers, t1; 2; 3;::: u.) Notation. If n P N, let σpnq denote the sum of all the divisors of n, σpnq “ d. d|n ¸ Definition. If n P N and σpnq “ 2n, then n is perfect. Euclid’s Elements contains the first recorded mathematical result regarding per­ fect numbers. Proposition 36 of Book IX of the Elements states: For some k ¡ 1 with 2k ´1 prime, 2k´1p2k ´1q is a perfect number.[8] Using the k values t2; 3; 5; 7u, we find that the first four perfect numbers are 6; 28; 496 and 8;128. Notice that σp6q “ 1 ` 2 ` 3 ` 6 “ 12 “ 2p6q. A similar calculation verifies the membership of 28; 496 and 8;128 in the set of perfect numbers which will be notated P. It should be noted that the cardinality of this set is neither known to be finite nor infinite. For future use, it will be beneficial to have some general notation for sets of natural numbers and their corresponding cardinalities. Notation. Let S Ď N with S “ |S| the cardinality of S. We will use the notation Spnq if we desire the subset S X r1; ns and Spnq for the cardinality of Spnq. As one might observe, for the set of perfect numbers that Euclid’s formula pro­ vides, there is a sizeable subset of the natural numbers that is unaccounted for. Making use of the divisor function, σpnq, it is observed that the set of natural numbers, N, can be partitioned into three sets of numbers, each possessing a dis­ tinct divisor sum characteristic. The characteristics are σpnq ¡ 2n, σpnq “ 2n and σpnq ă 2n. We shall call these subsets the set of abundant numbers, per­ fect numbers and deficient numbers, respectively. We will use the notation Λ for 2 the abundant numbers and D for the deficient. Note that Λ Y P Y D “ N and Λpnq Y Ppnq Y Dpnq “ Npnq. To assert that D is a non-empty set that is infinite, consider the fact that for any prime number p, σppq “ p ` 1 ă 2p and hence D contains the infinite subset P, the set of prime numbers. To assert that Λ is an infinite set, consider that for any positive multiple mn of n P P where m ¡ 1, σpmnq ¡ 2n. (We know that σpnq “ d “ 2n. Regardless d|n ¸ of what factors n and m have in common, we have that σpmnq ¥ m ¨ d ` 1 ¥ d|n mp2nq ` 1 ¡ 2mn.) ¸ 1.2 Density in Mathematics In mathematics, the density of a subset of numbers contained in a larger set is de­ fined to be the ratio of the cardinality of the subset to the cardinality of the larger set if this ratio exists. It should be clear that the density of any subset of a set will have a value in the interval r0; 1s. In the case that both the subset and parent set are finite, the calculation is quite elementary. For example, one could ask what is the density of numbers that are even in the set t1; 2; 3; 4; 5; 6; 7; 8; 9; 10u? The elements of the set that are even are t2; 4; 6; 8; 10u and hence the answer to the question is exactly determined by the calculation: |t2; 4; 6; 8; 10u| 5 ““ 0:5: |t1; 2; 3; 4; 5; 6; 7; 8; 9; 10u| 10 In the case where the two sets are not finite, the ratio of cardinalities is in indeterminate form and determining the density is not elementary, so we need a 3 new definition for the density of a subset and as it pertains to this paper, a subset of natural numbers. To aid in the calculation process, let’s develop some working notation to handle potentially infinite sets. Notation. Let S Ď N. Denote the natural density of S by dpSq, where dpSq “ Spnq lim , provided the limit exists. nÑ8 n With this notation, we can determine the natural density of any subset, S, of natural Spnq Spnq numbers, provided dpSq “ dpSq where dpSq “ lim and dpSq “ lim .
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