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Brun's 1920 Theorem on Goldbach's Conjecture Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 8-2018 Brun's 1920 Theorem on Goldbach's Conjecture James A. Farrugia Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Mathematics Commons Recommended Citation Farrugia, James A., "Brun's 1920 Theorem on Goldbach's Conjecture" (2018). All Graduate Theses and Dissertations. 7153. https://digitalcommons.usu.edu/etd/7153 This Thesis is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. BRUN’S 1920 THEOREM ON GOLDBACH’S CONJECTURE by James A. Farrugia A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mathematics Approved: David E. Brown, Ph.D. James S. Cangelosi, Ph.D Major Professor Committee Member Dariusz M. Wilczynski, Ph.D. Mark R. McLellan, Ph.D. Committee Member Vice President for Research Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2018 ii Copyright © James A. Farrugia 2018 All Rights Reserved iii ABSTRACT Brun’s 1920 Theorem on Goldbach’s Conjecture by James A. Farrugia, Master of Science Utah State University, 2018 Major Professor: Dr. David E. Brown Department: Mathematics and Statistics One form of Goldbach’s Conjecture asserts that every even integer greater than 4 is the sum of two odd primes. In 1920 Viggo Brun proved that every sufficiently large even number can be written as the sum of two numbers, each having at most nine prime factors. This thesis explains the overarching principles governing the intricate arguments Brun used to prove his result. Though there do exist accounts of Brun’s methods, those accounts seem to miss the forest for the trees. In contrast, this thesis explains the relatively simple structure underlying Brun’s arguments, deliberately avoiding most of his elaborate machinery and idiosyncratic notation. For further details, the curious reader is referred to Brun’s original paper (in French). Brun constructs two main sieves. For each of these sieves, he establishes a lower bound for the number of elements N that fall through the sieve. Then, he uses addi- tional results by Stirling and Mertens, along with innovative algebraic manipulations to construct improved lower bounds for N. Subsequently, Brun applies his earlier results, mutatis mutandis, to Merlin’s double sieve and obtains the result that allows him to prove his theorem on Goldbach’s Conjecture. In distilled form, Brun’s arguments run as follows: start with a lower bound of the form N > M − R, which this thesis calls “the Fundamental Inequality.” Then, bound M below by A and bound R above by B, to show that M > A − B. Thus, A − B is the improved lower bound for N. (95 pages) iv PUBLIC ABSTRACT Brun’s 1920 Theorem on Goldbach’s Conjecture James A. Farrugia One form of Goldbach’s Conjecture asserts that every even integer greater than 4 is the sum of two odd primes. In 1920 Viggo Brun proved that every sufficiently large even number can be written as the sum of two numbers, each having at most nine prime factors. This thesis explains the overarching principles governing the intricate arguments Brun used to prove his result. Though there do exist accounts of Brun’s methods, those accounts seem to miss the forest for the trees. In contrast, this thesis explains the relatively simple structure underlying Brun’s arguments, deliberately avoiding most of his elaborate machinery and idiosyncratic notation. For further details, the curious reader is referred to Brun’s original paper (in French). v ACKNOWLEDGMENTS Thanks to everyone who helped. James A. Farrugia vi Contents Page ABSTRACT.................................... iii PUBLIC ABSTRACT.............................. iv ACKNOWLEDGMENTS............................ v LIST OF FIGURES...............................viii 1 INTRODUCTION............................... 1 1.1 Overview....................................1 1.1.1 What Brun Did to Achieve his Result................1 1.1.2 Preview of the Fundamental Inequality: N > M − R ........2 1.1.3 Plan...................................2 1.2 The Sieve of Eratosthenes...........................3 1.3 The Sieve of Eratosthenes-Legendre.....................6 1.4 The Legendre Formula.............................8 1.4.1 Chief Limitation of the Legendre Formula.............. 10 1.4.1.1 What is the Error Term in the Legendre Formula?.... 11 1.4.1.2 What is Meant by Saying “the Error Term is Too Large”? 12 1.4.1.3 Why is a Large Error Term Problematic?......... 15 1.4.2 Of What Use, then, is the Legendre Formula?............ 16 1.5 The “Best” Lower Bound for N ........................ 17 1.6 Revisiting the Fundamental Inequality: N > M − R ............ 19 1.7 Sieve Integrity when Truncating the Legendre Formula........... 20 1.8 Why Some Composites Fall Through Brun’s Sieves............. 22 1.9 Brun’s Use of Merlin’s Double Sieve..................... 23 1.10 Contextual View of Brun’s Approach..................... 25 2 GOLDBACH’S CONJECTURE AND RELATED RESULTS . 28 2.1 Goldbach’s Conjecture............................. 28 2.2 Some Evidence that Goldbach’s Conjecture is True............. 29 2.2.1 Empirical Evidence........................... 29 2.2.2 The Prime Number Theorem and Heuristics............. 32 2.2.3 Sieve Methods............................. 35 2.2.4 The Circle Method........................... 36 2.2.5 Evidence Based on the Density of Primes.............. 36 2.2.6 Lower Bounds, Upper Bounds, and Asymptotic Estimates..... 36 2.3 Significance: Goldbach’s Conjecture, Sieve Methods, Brun’s Work.... 36 vii 2.3.1 Significance of Goldbach’s Conjecture................ 36 2.3.2 Significance of Sieve Methods..................... 38 2.3.3 Significance of Brun’s Sieve Methods................. 38 3 BRUN’S THEOREM ON GOLDBACH’S CONJECTURE . 40 3.1 Overview of “Le Crible d’Eratosth`eneet le Th´eor`emede Goldbach”... 40 3.2 Different Lower Bounds and Brun’s Generalized Notation......... 42 3.2.1 Different Lower Bounds for the Legendre Formula......... 42 3.2.1.1 Searching for Improved Lower Bounds........... 42 3.2.1.2 The Default (“Worst”) Lower Bound, Revisited..... 43 3.2.1.3 Discarding Terms in “Small Print”............. 44 3.2.1.4 Discarding Terms “to the Right of Vertical Lines”.... 46 3.2.2 Brun’s Notation in §2 for his Generalized Sieve........... 48 3.3 Discarding Terms in “Small Print”...................... 49 3.3.1 Algorithm for Discarding Terms in “Small Print”.......... 50 3.3.2 Brun’s Sieve in §3 Does Give a Lower Bound for N ......... 51 3.3.3 Stirling’s Formula is Used to Bound Part of the Expansion.... 53 3.3.4 Bounding the Main Term....................... 54 3.3.5 Bounding the Remainder Term.................... 55 3.3.6 Combining Results Yields a New Lower Bound for N ....... 56 3.3.7 Problem: the Improved Lower Bound Could Still be Negative... 56 3.4 Discarding Terms “to the Right of Vertical Lines”.............. 56 3.4.1 Algorithm for Discarding Terms “to the Right of Vertical Lines”. 57 3.4.1.1 The Basic Idea Behind the Algorithm........... 57 3.4.1.2 Brun’s Notation for the Algorithm............. 59 3.4.2 Brun’s Sieve in §4 Does Give a Lower Bound for N ......... 63 3.4.2.1 Examples Illustrating Key Formulas from Brun’s §2... 64 3.4.2.2 Deriving Brun’s Formula (2) from his Formula (1).... 64 3.4.2.3 From Brun’s Formula (2) to Formula (3)......... 65 3.4.2.4 The “N-terms” in Formula (3)............... 67 3.4.2.5 An Example Illustrating Brun’s Formula (13)....... 69 3.4.2.6 Brun’s Formula (14) follows from his Formula (13).... 71 3.4.3 Overview of Brun’s Steps in §4.................... 72 3.4.4 Assumptions Used by Brun for his Sieve in §4............ 73 3.4.5 The Values of Particular Determinable Numbers is Never at Issue 76 3.4.6 Bounding the Main Term....................... 76 3.4.7 Bounding the Remainder Term.................... 77 3.4.8 Combining and Extending Results.................. 78 3.5 Brun Proves his Theorem on Goldbach’s Conjecture............ 79 3.5.1 Overview................................ 79 3.5.2 Initial Lower Bound for the Double Sieve.............. 79 3.5.3 Bounding the Main Term and the Remainder............ 80 3.5.4 Combining and Extending Results.................. 81 3.6 Summary.................................... 82 REFERENCES.................................. 83 viii LIST OF FIGURES Figure Page 1.2.0.1 Primes and 1 fall through the sieve of Eratosthenes.............5 1.4.0.1 Finding N(1, 37, 2, 3, 5)............................8 1.7.0.1 Finding N(1, 37, 2, 3, 5)............................ 20 1.8.0.1 Finding N(1, 37, 2, 3): the 2-primes 25 and 35 fall through......... 22 1.9.0.1 Brun’s way of showing a sieve......................... 23 1.9.0.2 Brun’s way of showing his double sieve.................... 24 1.9.0.3 Double sieve starting with sequence of consecutive odd numbers..... 25 1.9.0.4 Double sieve allowing the 2-prime 25 to fall through............ 25 2.2.1.1 r(n): number of distinct representations of n as sum of two odd primes. 30 2.2.1.2 As n increases, r(n) reveals its signature pattern.............. 30 2.2.2.1 Consider r(50): with “matching pairs of primes” connected........ 32 n 2.2.2.2 Red line (log n)2 tracks the data but is too high for a lower bound..... 34 n 2.2.2.3 Red line (log n)2 tracks the data but is too high for a lower bound..... 34 √ 2 n 2.2.2.4 Red line 2 (log n)2 − 4, a plausible lower bound for r(n).........
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