In this note, we utilize techniques from discrete mathematics to develop first an inequality, and then second a counting formula that is connected to Goldbach\u27s conjecture. In order to do this, we introduce the notion of a Relative Goldbach Partition

Hutchinson, Houston

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Решетников, Сергей Васильевич

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Приоритеты развития политических исследований на кафедре политологии БГУ

Furman University

Furman UniversityElectronic Journal of Undergraduate MathematicsVolume 15, 1 – 4, 2011RELATIVE GOLDBACH PARTITIONS AND GOLDBACH’SCONJECTUREHOUSTON HUTCHINSONAbstract. In this note, we utilize techniques from discrete mathematics todevelop first an inequality, and then second a counting formula that is con-nected to Goldbach’s conjecture. In order to do this, we introduce the notionof a Relative Goldbach Partition.1. IntroductionThe allure of Goldbach’s conjecture has captured the interest of many. It hasappeared in the story lines of television shows, movies, and recent novels such asUncle Petros and Goldbach’s Conjecture by Apostolos Doxiodis [1]. The conjectureis often introduced to undergraduates in their first or second year in discrete math-ematics texts as in [2]. The conjecture says that every even integer greater than2 can be written as the sum of two prime numbers. In this note, we connect thepigeon-hole principle to Goldbach’s conjecture for certain even integers and thendevelop an interesting counting formula utilizing what we term Relative GoldbachPartitions. All of our results are accessible to students in a first course in discretemathematics.2. A Pigeon-Hole ArgumentWe denote the Euler totient of a positive integer n by φ(n). Recall that thisgives the number of positive integers less than n that are relatively prime to n. Wedefine ρ(n) for a positive, even integer n to be the number of primes that are lessthan n but are not among the prime factors of n. We claim that if ρ(n) > φ(n)/2,then n can be written as the sum of two distinct primes. To see this, observe thatan even integer n can be written as the sum of two positive numbers in n/2 ways:1 + n− 1, 2 + n− 2, . . . , n/2 + n/2.If the first summand is relatively prime to n, then so is the second. It follows thatφ(n)/2 of the sums consist of two numbers that are relatively prime to n. We thinkof ρ(n) as pigeons and φ(n)/2 as pigeon-holes. Thus if ρ(n) > φ(n)/2, then n can bewritten as a sum of two distinct primes. As an example, we apply our pigeon-holeargument to the number 22. Since ρ(22) = 6 and φ(22)/2 = 5, we are guaranteedthat a sum of distinct primes exists. Note that 22 = 11 + 11, 3 + 19, and 5 + 17.Unfortunately, the inequality ρ(n) > φ(n)/2 fails more often than it holds. InFigure 1, we compare a graph of ρ(n) with φ(n)/2 for even values of n out to 1000.Received by the editors July 11, 2011.2000 Mathematics Subject Classification. 11P32.Key words and phrases. Goldbach’s Conjecture, Goldbach Partitions.12 HOUSTON HUTCHINSON0 100 200 300 400 500 600 700 800 900 1000050100150200250  Figure 1. φ(n)/2 in blue compared to ρ(n) for even n out to 1000The blue dots below the red line in Figure 1 represent values of n for which we mayapply our pigeon-hole argument.3. Relative Goldbach PartitionsA pair of prime numbers that sum to an even integer n is called a GoldbachPartition. If Goldbach’s conjecture holds, then every even integer n greater than 2has at least one Goldbach Partition. For any positive integer n, we define a RelativeGoldbach Partition (RGP) to be a pair of positive numbers that sum to n such thatthe numbers are relatively prime to n and the summands are not themselves primenumbers. If n− 1 is not prime, then n has at least one RGP namely {1, n− 1}. Itis interesting to note that the first number that has two RGP’s is 34. The number34 can be written as 1 + 33 and 9 + 25. For an integer n, we denote the numberof unordered RGP’s by R(n). The sequence R(n) is a recent addition to the onlineencyclopedia of integer sequences. It matches A185279 while R(2n) is the sameas A141095 (see [3] and [4]). We also denote the number of unordered GoldbachPartitions consisting of distinct primes by G(n). In Figure 2, we see the graph ofR(n) and we invite the reader to compare it to the well known graph of Goldbach’scomet.4. A Counting FormulaWe next derive a formula for G(n) utilizing the functions ρ(n), R(n) and φ(n).We claim that for an even integer n,G(n) = ρ(n) +R(n)− φ(n)/2.To see this, consider the φ(n) numbers that are relatively prime to n. Pairing thesenumbers into sums of n yields three possibilities. The sum contributes to the valueof G(n), the sum contributes to the value of R(n), or one of the summands is primeand the other is a number that is relatively prime to n. We call this a type 3 sum.There are two ways to count the number of type 3 sums. The first way is to takeall of the φ(n) many numbers, subtract all primes that remain, and delete all pairsRELATIVE GOLDBACH PARTITIONS AND GOLDBACH’S CONJECTURE 30 500 1000 1500 2000 2500050100150200250300Figure 2. The comet corresponding to R(n)that contribute to R(n). Thus the number of type 3 sums is equal toφ(n)− ρ(n)− 2R(n).The second way to count the number of type 3 sums is to do the following. Startonly with the primes that contribute to the value of ρ(n). If one subtracts all ofthe primes that are a member of a distinct Goldbach Partition, then what is left isagain equal to the number of type 3 sums. This is the same asρ(n)− 2G(n).Equating the two expressions that represent type 3 sums and solving for G(n) givesthe desired result.Our counting formula implies that if ρ(n) +R(n) > φ(n)/2, then G(n) > 0. Thelatter inequality appears to hold for all integers greater than 6. This means that onecould rephrase Goldbach’s conjecture to say that every even integer greater than 6can be written as the sum of two distinct primes. Thus if Goldbach’s conjectureis false, then there exists an integer n such that ρ(n) + R(n) = φ(n)/2. Based onthe computational efforts of many (see e.g. [5]), such an n would no doubt be verylarge!————————————————————References.1. A. Doxiodis, Uncle Petros and Goldbach’s Conjecture, Bloomsbury 2001.2. E. Goodaire and M. Parmenter, , Discrete Mathematics with Graph Theory,Pearson Prentice Hall, 2006.3. OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences,http://oeis.org/A185279.4. OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences,http://oeis.org/A141095.5. J. Richstein, Verifying the Goldbach Conjecture up to 4 · 1014, Mathematics ofComputation, 70 (2001), 1745-1749.Abilene Christian UniversityE-mail address: houston.hutchinson@googlemail.com4 HOUSTON HUTCHINSONSponsor: Jason Holland, Assoc. Professor, Abilene Christian UniversityE-mail address: hollandj@acu.edu

Relative Goldbach Partitions and Goldbach\u27s Conjecture

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Relative Goldbach Partitions and Goldbach\u27s Conjecture

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