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# Goldbach Conjecture

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• 1. Maths What different approaches have been applied in attempts to mathematically prove and verify that Goldbach’s binary and ternary conjectures are true? Name: Anil Prashar Word count: 3718 words Abstract Word Count: 211 words
• 2. IB Extended Essay: Maths Anil Prashar Abstract This essay attempts to compare and analyse some of the successful, classical methods, as well as some of the new more interesting approaches that have been applied in the attempt to prove Goldbach’s conjecture over time. There have been various different approaches from mathematicians of varied calibres, each either furthering the conjecture or helping to change the way the conjecture is viewed. I attempt to consider and compare different approaches not strictly with the use of rigorous proof as most ideas are presented in maths. Instead I compare the key points of each idea and method, whilst considering it in terms of the initial conjecture. The first real attempts to analyse Goldbach’s conjecture were in the 1900’s. In this essay I attempt to track the progress of number theory, by looking at methods that have been applied in the last century. The question is of significance as it considers two fairly fundamental ideas in mathematics, in the sense that; many amateurs and mathematicians find the conjecture easy to understand. I will therefore be approaching the conjecture from the position of an amateur, with the knowledge of a college student. Understandably, this makes some ideas inaccessible to me, but this essay attempts to understand the key points behind theorems and ideas involved. 2
• 3. IB Extended Essay: Maths Anil Prashar Table of Contents I) Introduction......................................................................page 3 II) The Binary and Ternary Conjectures; Linked..............page 4 III) Proof by Probability.........................................................page 6 IV) Schnirelmann’s Theorem…………………………..….page 8 V) The Smallest Partition………………………………....page 11 VI) Sieve Theory………………………………………...….page 14 VII) Graphical methods and Vinogradov’s Theorem…….page 18 VIII) Conclusion………………………………………...……page 21 IX) Bibliography………………………………………..…..page 22 3
• 4. IB Extended Essay: Maths Anil Prashar I) Introduction In 1742 the amateur mathematician, Christian Goldbach, wrote a letter to the infamous Leonhard Euler. The letter suggested a possibility regarding number theory; it would later come to be known as ‘Goldbach’s Conjecture’. This conjecture stated two major ideas: 1. Every even integer ≥ 6 can be expressed as the sum of two positive, prime numbers; often named Goldbach’s binary (strong) conjecture. 2. Every odd integer ≥ 9 can be expressed as the sum of three positive, prime numbers; often named Goldbach’s ternary (weak) conjecture.1 Part 1 means that 6, 8, 10, 12 etc. (all even integers) can be displayed as the sum of two primes. Examples as to why this may be true are shown below: 6 = 3+3 8 = 3+5 10 = 3 + 7 = 5 + 5 Part 2 states that 9, 11, 13, 15 etc. (all odd integers) can be 12 = 5 + 7 displayed as the sum of three primes. Examples are shown below: 9 = 2+ 2+5 = 3+3+3 11 = 2 + 2 + 7 = 3 + 3 + 5 13 = 3 + 3 + 7 15 = 2 + 2 + 11 = 3 + 5 + 7 Euler agreed with Goldbach’s findings, but stated that he was unable to find the proof himself. The binary and ternary conjectures remain unproved to this day. The conjecture is, perhaps, particularly interesting as there have been numerous approaches to the Goldbach conjecture based on the state of number theory. By exploring how different methods have been applied over time, one can see the development of number theory in its attempts to prove conjectures. II) The Binary and Ternary Conjectures; Linked 1 Wang, Yuan. The Goldbach Conjecture. London: World Scientific Publishing Company; 2 Sub Edition, 2003. 4
• 5. IB Extended Essay: Maths Anil Prashar Let N = a sufficiently large positive, even integer p1 and p2 = two prime numbers (may be identical) If N = p1 + p 2 (i.e. If Goldbach’s binary conjecture is true) Then N + 3 = p1 + p 2 + 3 ∴ N + 3 = p1 + p 2 + p 3 2 This therefore suggests that if Goldbach’s binary conjecture is correct then the ternary conjecture is proved by default. This is because an even number plus three is odd. In addition to this an even number plus any other prime number (excluding 2) is always odd, and therefore p3 could just as easily have represented 5, 7, 11, 13 etc; thus creating further combinations for the composition of an odd number. I also consider this idea in a different light. For example if the ternary conjecture were able to be proved on its own merit, then even numbers could be shown to be the sum of 4 primes as opposed to 2 (shown below). Let O = a sufficiently large positive, odd integer p1, p2 and p3 = three non-identical prime numbers If O = p1 + p 2 + p 3 (i.e. if Goldbach’s ternary conjecture is true) Then O + 3 = p1 + p 2 + p 3 + 3 ∴ O + 3 = p1 + p 2 + p 3 + p 4 2 Shi, Kaida. "A New Method to Prove Goldbach Conjecture, Twin Primes Conjectures and Other Two Propositions." Zhejiang Ocean University, 2000: 2. 5
• 6. IB Extended Essay: Maths Anil Prashar As any odd number plus another odd number is always even, then all even numbers could be shown to be the sum of 4 primes. However, both of these ideas are dependent on the massive assumption that either the binary or ternary conjectures are already true. As the binary automatically proves the ternary it is only natural that it has been subject to greater attention i.e. it has had more attempted proofs. It is now time to consider the conjectures themselves. The reason as to why the integers had to be ‘sufficiently large’ is explained later in part VI [Sieve Theory]. 6
• 7. IB Extended Essay: Maths Anil Prashar III) Proof by Probability Some mathematicians conjecture that if the binary conjecture is relatively simple and easy to understand, then perhaps the proof shares a similar simplicity. This has led to some people accepting Goldbach’s theory under the heuristic method of probability. Probability is generally an unaccepted method of proving a conjecture in maths; however it seems to be one of the methods used to enhance the idea that Goldbach’s conjecture is correct. The process of thought is as follows: As the even integer that is being considered increases, the number of partitions also increases: 8 = 3+5 18 = 5 + 13 = 7 + 11 24 = 5 + 19 = 7 + 17 = 11 + 13 This is perhaps further clarification that as we continue to verify larger even numbers are the sum of two primes our chances of finding one that isn’t the sum of two primes diminishes. Thus far the Goldbach conjecture has been verified by computers to be correct until 1.2 × 1018 suggesting that now a pattern or proof is to be discerned3 . A particularly distinct approach has been used by the mathematician, Mark Herkommer; he begins by examining the probability of prime pairs occurring in the partition of 100 based on intervals of size ten4: n = 100 probability Probability Interval % prime Matching Interval % prime prime pair neither prime pair 0 - 10 60 90 – 100 20 0.12 0.88 10 - 20 80 80 – 90 40 0.32 0.68 20 - 30 40 70 – 80 60 0.24 0.76 30 - 40 40 60 – 70 40 0.16 0.84 40 - 50 60 50 – 60 40 0.24 0.76 • % prime: Shows the percentage of the odd numbers in the interval that are prime. 3 T., Oliveira e Silva. Goldbach Conjecture Verification. July 14, 2008. http://www.ieeta.pt/~tos/goldbach.html (accessed July 1, 2009). 4 Herkommer, Mark. Goldbach Conjecture Research. May 24, 2004. http://www.petrospec- technologies.com/Herkommer/goldbach.htm (accessed June 30, 2009). 7
• 8. IB Extended Essay: Maths Anil Prashar The probabilities of a prime pair summing to make 100 are calculated by multiplying the percentage of prime numbers in the ‘Interval’ and the percentage of prime numbers in the ‘Matching Interval’. To calculate the probability of a prime number partition being possible for 100, you multiply all the probabilities of the event not happening. Then as there are only two possibilities; either the event happening or not happening. The probabilities must sum to make 1 and ‘1 – the total probability of the event not happening’, in this case, gives you the probability 0.70967. If this process is then repeated on larger values, then the probability can be found to increase as shown below5: n probability 1000 0.996208045988 2000 0.999838315754 3000 0.999999069064 4000 0.999999693974 5000 0.999999983603 6000 0.999999999995 7000 0.999999999875 8000 0.999999999978 9000 1.000000000000 10000 1.000000000000 However, as the author of the method recognises, the larger probabilities occur when smaller intervals are used. He also appreciates that there may be the one number that is not noticeable through such an imprecise method. The main point of this idea is to recognise that Goldbach’s conjecture, in terms of probability, is most likely true. IV) Schnirelmann’s Theorem 5 ibid. 8
• 9. IB Extended Essay: Maths Anil Prashar ‘There exists a positive integer “s” such that every sufficiently large integer is the sum of at most “s” primes. It follows that there exists a positive integer s0 ≥ s such that every integer > 1 is a sum of at most s primes. The smallest proven value of s 0 is known as the Schnirelmann constant.’6 The magnitude of this theorem meant that Goldbach’s conjecture was finally to be brought into the finite realms of mathematical investigation. The proof is based on Mann’s theorem. Mann’s theorem states that if ‘A and B are sets of integers each containing 0’, then: ‘ σ ( A ⊕ B) ≥ min{1,σ ( A) + σ ( B )} ’7 • σ represents the Schnirelmann density, defined as the ‘greatest lower bound of the S ( n) fractions where S (n) is the number of terms in the set ≤ n ’8. n • {n ≥ 1, n ∈ Q} where Q represents a set. e.g. Set: {0, 2, 4, 6, 8} 5 When n = 8 , S (n) = 5 {σ < } 8 In this situation, trial and error was used to calculate the Schnirelmann density; the S ( n) greatest lower bound is the smallest value of which in this case was < 5 8 . n • Greatest Lower Bound: If it is given the value c in set Q , then c ≤ x, ∀x ∈ Q . This could otherwise be seen as 0 ≤ c ≤ smallest value of x . • A ⊕ B = {a + b, a ∈ A, b ∈ B} represents the direct sum, where each element of set A is added to every element of set B, any repeated values are discarded. • min( A, B ) represents the smallest value in the set A or B, depending on which one is smaller. 6 O'Bryant, Kevin. Schnirelmann's Theorem. http://mathworld.wolfram.com/SchnirelmannsTheorem.html (accessed July 2, 2009). 7 O'Bryant, Kevin. Mann's Theorem. http://mathworld.wolfram.com/MannsTheorem.html (accessed July 2, 2009). 8 Weisstein, Eric W. Schnirelmann Density. http://mathworld.wolfram.com/SchnirelmannDensity.html (accessed July 2, 2009). 9
• 10. IB Extended Essay: Maths Anil Prashar So to simplify the previous points and bring it back into the context of Schnirelmann’s Theorem, Mann’s theorem states that the greatest lower bound of the sum of two sets is greater than or equal to the greatest lower bound of set A plus the greatest lower bound of set B. In terms of Schnirelmann’s theorem, one very important point to remember is that 0 ∈ A ∩ B , as mentioned earlier. Let P = {0,1} ∪ { p} = {0,1, 2,3, 5, 7, ...} Where p is all prime numbers Q= P+P = {0,1,2,3,5,7,1,2,3,4,6,8,2,3,4,5,7,9,3,4,5,6,8,10,5,6,7,8,10,12,7,8,9,10,12,14} By using the direct sum method and discarding all repeated values, you get: Q = {0,1,2,3,4,5,6,7,8,9,10,12,14...} It can be shown using the inclusion-exclusion principle9 that: σ ( P) = 0 But σ (Q) > 0 S ( n) Now if we consider that σ (Q) is the greatest lower bound of , and is > 0 , then the n following method applies: P = {0,1, 3, 5, 7,11...} σ ( P) = 0 Q = {0,1, 2, 3, 4, 5,6,8,10,12,14...} σ (Q) > 0 Remember that as σ ( P + P) ≥ min{1,σ ( P ) + σ ( P )} , the maximum value that σ ( P + P ) can ever take, depending on how many times P is added, will be 1. Therefore if this process is then repeated using Q + Q , then Q + Q + Q + Q , this leads to: σ (kQ) = kσ (Q) = 1 (This is considered an acceptable rule) 9 Inclusion-Exclusion Principle. June 26, 2009. http://en.wikipedia.org/wiki/Inclusion- exclusion_principle (accessed July 1, 2009). 10
• 11. IB Extended Essay: Maths Anil Prashar 1 10 ∴k = σ (Q) This is because eventually S (n) = n , therefore the density = 1. This is clearly noticeable in set Q, as it has many more numbers, causing its Schnirelmann density to increase. It is also only possible for sets to have Schnirelmann density = 1, if and only S (n) if the set contains all positive integers, because then = 1 at every value. n Therefore if the process is repeated k times on the initial prime number set, then k can be used to deduce the maximum amount of prime numbers that sum to make any positive integer. This process is intensely intricate and is carried out through the use of various set theory principles. With it Schnirelmann was able to find the number s0, which represented the minimum amount of prime numbers required to represent the sum of each number. Schnirelmann initially found the constant was 300000 prime numbers11. This was improved over time, as shown below12: s0 Author 15 Deshouillers (1973) 9 11 Klimov et al. (1972) 5 55 Klimov (1975) 27 Vaughan (1977) 26 Deshouillers (1977) 19 Riesel and Vaughan (1983) 7 Ramaré (1995) 10 Schnirelmann's Theorem. June 27, 2009. http://en.wikipedia.org/wiki/Schnirelmann_constant#Schnirelmann.27s_theorem (accessed July 1, 2009). 11 Hofstadter, Douglas R. Goedel, Escher, Bach; an Eternal Golden Braid. London: Penguin Books Ltd., 2000. 12 Weisstein, Eric W. Schnirelmann Constant. http://mathworld.wolfram.com/SchnirelmannConstant.html (accessed July 2, 2009). 11
• 12. IB Extended Essay: Maths Anil Prashar As can be seen from this table, Ramaré has developed the idea furthest with his findings, using Schnirelmann’s initial theorem and sieve theory to prove that at most 7 prime numbers are required to represent all numbers. Although this is close to Goldbach’s initial conjectures, it is not quite there yet. V) The Smallest Partition When considering the Goldbach conjecture, one large difficulty is the fact that there are numerous partitions for each number making patterns more difficult to discern. In this situation some mathematicians have approached the numbers looking at where the smallest prime number partition exists. The results for n<1000000000 are as below13: n g(n) n - g(n) g(n) / n 6 3 3 0.500000000 12 5 7 0.416666667 30 7 23 0.233333333 98 19 79 0.193877551 220 23 197 0.104545455 308 31 277 0.100649351 556 47 509 0.084532374 992 73 919 0.073588710 2642 103 2539 0.038985617 5372 139 5233 0.025874907 7426 173 7253 0.023296526 43532 211 43321 0.004847009 54244 233 54011 0.004295406 63274 293 62981 0.004630654 113672 313 113359 0.002753536 128168 331 127837 0.002582548 194428 359 194069 0.001846442 194470 383 194087 0.001969455 413572 389 413183 0.000940586 503222 523 502699 0.001039303 1077422 601 1076821 0.000557813 3526958 727 3526231 0.000206127 3807404 751 3806653 0.000197247 10759922 829 10759093 0.000077045 24106882 929 24105953 0.000038537 27789878 997 27788881 0.000035876 37998938 1039 37997899 0.000027343 113632822 1163 113631659 0.000010235 187852862 1321 187851541 0.000007032 13 (Herkommer 2004) 12
• 13. IB Extended Essay: Maths Anil Prashar 335070838 1427 335069411 0.000004259 419911924 1583 419910341 0.000003770 721013438 1789 721011649 0.000002481 n = a random even integer between 0 and 1000000000 g(n) = the smallest prime number that sums with another prime number to make n = Due to simple addition this becomes the largest prime number that sums to make n with another prime. = An interesting column that shows the ratio between n and the smallest partition value. One evident observation in the column is that the value of this is ratio is decreasing. This idea is graphically portrayed below14: Where X-axis = log10n [used to help show numbers on a closer scale] Y-axis = g(n) 14 ibid. 13
• 14. IB Extended Essay: Maths Anil Prashar As is evident from the graph, there is a clear exponential incline that can be discerned from the graph. Below I have added a curve of best fit to the graph to analyse it more efficiently: Perhaps the most striking detail about the curve is that most of the points do not lie on it. This therefore means that there isn’t a consistent increase in the ratio of the smallest partition to the initial number; in essence nullifying the method. It appears that all partitions need to be considered if the conjecture is to have a consistent proof. 14
• 15. IB Extended Essay: Maths Anil Prashar VI) Sieve Theory You may be aware of the sieve of Eratosthenes, a method used to find prime numbers by cancelling out multiples of a number as you continue to move across a number line. In 1915, the mathematician Viggo Brun developed a new type of sieve, known today as Brun’s sieve. Brun’s sieve is an estimation method15, where the sizes of ‘sifted sets’ are estimated based on congruences between groups based on set conditions. A congruence ‘on a set X determines a partition of the set X to which it corresponds’16. In simpler terms this means that when a certain condition is applied to a set, a congruence is what determines each of the subsets created by the condition. A simple example might be where a large set (all numbers in this case) is taken, and all even numbers are taken and placed in a subset; the congruence is the determination that one subset is all even numbers. The odd numbers are also placed in another subset. Now if I took specific elements from the original set (all numbers), in this case I shall take 7 and 16, they are only considered equivalent based on the subset that they are placed in. As 7 is odd and 16 is even, they are not considered equivalent. However if I had taken 10 and 16 they would have been considered equivalent based on the fact that they are in the same subset. To discuss this concept further, it is first necessary to go back to Euler and his works. One of the key observations that Euler had made in his works was that the sum of the reciprocals of all prime numbers diverges17. It was Brun who made an even more interesting observation however. By using the observation that: 18 ln ln x 2 π 2 ( x ) = O( x ( )) ln x • π 2( x) is defined as { p = x p + 2 = q} , where p and q are both prime. The mathematical term for these types of numbers is ‘twin primes’. 15 Brun Sieve. May 19, 2009. http://en.wikipedia.org/wiki/Brun_sieve (accessed June 27, 2009). 16 Eccles, Peter J. An Introduction to Mathematical Reasoning; numbers, sets and functions. Cambridge: Cambridge University Press, 2001. 17 Caldwell, Chris K. There are infinity many primes, but how big of an infinity? http://primes.utm.edu/ infinity.shtml (accessed June 30, 2009). 18 Charles, Denis Xavier. "Sieve Methods." 2000: 35. 15
• 16. IB Extended Essay: Maths Anil Prashar • The ‘O’ function is the Landau notation. This can be described as follows: As x → ∞ f ( x) = O( g ( x)) But this is only if for a large value of x there exists positive real integers for M and x 0 such that: f ( x) ≤ M g ( x) For all values of x Where M is a constant19 This basically means that the Landau notation is used to help express f (x ) as x tends to infinity in the form of a simplified g (x) . For example: Let f ( x) = 3 x 3 − 4 x 2 + 7 Now we want to consider the behaviour of f (x ) as x → ∞ . It is obvious that 3x 3 will grow the fastest, as it has the highest derivative ( 9x 2 ). f ( x) = O( x 3 ) [Remember x → ∞ ] Now this needs to be in the same form as f ( x) ≤ M g ( x) , where M and x 0 are positive, real integers. Let x0 = 1 3 x 3 − 4 x 2 + 7 ≤ 3x 3 + 4 x 2 + 7 [Modulus has been taken] 3 x 3 − 4 x 2 + 7 ≤ 3x 3 + 4 x 3 + 7 x 3 [ x0 = 1 ] 3 x 3 − 4 x 2 + 7 ≤ 14 x 3 [Collect like terms] Thus to use the original form: 3 x 3 − 4 x 2 + 7 ≤ 14 x 3 For all x > x 0 19 Big O Notation. June 28, 2009. http://en.wikipedia.org/wiki/Big_O_notation (accessed July 1, 2009). 16
• 17. IB Extended Essay: Maths Anil Prashar In this particular circumstance, as the twin primes tend to infinity, they have been able to be expressed in a simpler form using Landau notation. It was with this form that Brun proved that the sum of all twin primes converges. As he found that the value for the sum of the reciprocals of all twin primes was around 1.90216054, a number now known as ‘Brun’s Constant’. Its finding meant that there were believed to be a finite amount of twin primes. In consideration of Goldbach’s conjecture, we are working with the assumption that there is an infinite amount of numbers. Therefore for Goldbach’s binary and ternary conjectures to be true there must be an infinite amount of prime numbers. It was proved true by Euler that there were an infinite amount of prime numbers20. However the suggestion that there are a finite amount of twin primes has possible implications for Goldbach’s conjecture, as certain numbers are only expressible as the sum of twin primes e.g. 3 + 5 = 8, 5 + 7 = 12 etc. On the other hand, going back to part III [Proof by Probability], larger numbers generally have more partitions, and therefore twin primes may be considered to have no relevance to Goldbach’s conjecture (except at early stages) e.g. 24 is the sum of the twin primes 11 and 13, but is also the sum of 5 and 19. Hardy also suggested a further link between twin primes and Goldbach’s conjecture by suggesting that the function G(N) [which represents the number of ways in which N can be written as the sum of two primes] was asymptotic to some function of the twin prime constant21. The twin primes constant is defined as: 1 ∏ (1 − ( p − 1) p >2 2 ) Where p is a prime number A solid link between Goldbach’s conjecture and twin primes was validated by Chen Jingrun in 1966, using an extremely long and rigorous proof; with the help of sieve 20 Caldwell, Chris K. Euclid's Proof of the Infinitude of Primes. http://primes.utm.edu/notes/proofs/infinite/euclids.html (accessed June 29, 2009). 21 Caldwell, Chris K. Goldbach's Conjecture. http://primes.utm.edu/glossary/xpage/GoldbachConjecture.html (accessed June 29, 2009). 17
• 18. IB Extended Essay: Maths Anil Prashar theory. Some very interesting and important points of consideration came from his proof22, where he deduced that all sufficiently large even numbers (represented as N) could be partitioned in two possible ways: 1) N = p1 + p 2 2) N = p1 + p 2 p 3 (Where p1, p2 and p3 are all prime numbers) Form 1) was mentioned in part II of this essay [The Binary and Ternary Conjectures; Linked]. This is an important observation, for Chen has perhaps come closest to proving the binary conjecture with his sieve method. It has already been discussed how Chen’s form 1) would prove the ternary conjecture by default. However form 2) recognises that some even numbers have been sorted as the sum of a prime and a semi-prime. A semi-prime is defined as the product of two prime numbers. Thus the conjecture still remains unproved. It is perhaps possible, using more refined methods of ‘sieving’ and estimation to finally prove Goldbach’s conjecture, yet such methods have not been attempted as of yet. 22 PrimeFan. Chen's Theorem. http://planetmath.org/encyclopedia/ChensTheorem.html (accessed June 27, 2009). 18
• 19. IB Extended Essay: Maths Anil Prashar VII) Graphical methods and Vinogradov’s Theorem An interesting method pursued by some mathematicians is the use of graphs23: The graph shown is known as ‘Goldbach’s Comet’, a few propositions that have been made in this essay become explicitly clear through the use of this graph. For example the number of partitions increases as the number being considered increases (Part III [Proof by Probability]). The main point of using a graph is based on its curving shape. It has been suggested that the proof may be dependent on some sort of asymptotic approach in order to help prove the conjecture. This approach was advocated by various renowned mathematicians in their attempt to tackle Goldbach’s conjecture e.g. Ramanujan, Hardy, Erdös etc. However, it was to be the Russian mathematician, Ivan Vinogradov, who would effectively find the method that has made one of the most important leaps in helping 23 (Herkommer 2004) 19
• 20. IB Extended Essay: Maths Anil Prashar to prove Goldbach’s ternary conjecture. He found that every ‘sufficiently large’ odd integer could be expressed as the sum of three primes. He did indeed use asymptotic analysis to give finite bounds for the number of partitions that could be made of odd integers as the sum of three primes. His theorem is as follows: Let A = a positive, real integer N = a sufficiently large odd integer p1, p2 and p3 = three primes that sum to make ‘N’ Then according to Vinogradov’s theorem: 24 • represents the function of N that is used to represent Vinogradov’s theorem. • G(N) represents the number of ways in which N can be partitioned as the sum of a certain amount of primes. • You may once more recognise the (O) Landau notation, used to show r(N) as N → ∞ . This is a quantity that is required in order to assess r(N) asymptotically (as it tends to infinity). And: r(N ) = ∑ [Λ( p1) Λ( p2) Λ( p3)] p1+ p 2 + p 3= N Where: • (n) {known as the Von Mangoldt function} = [If where p is a prime and a is an integer ≥ 1] 25 0 [If ] Therefore: r(N ) = ∑ [Λ ( p1) Λ ( p2) Λ ( p3)] p1+ p 2 + p 3= N 24 Vinogradov's Theorem. June 24, 2009. http://en.wikipedia.org/wiki/Vinogradov%27s_theorem (accessed June 27, 2009). 25 Von Mangoldt function. May 2, 2009. http://en.wikipedia.org/wiki/Von_Mangoldt_function (accessed June 30, 2009). 20
• 21. IB Extended Essay: Maths Anil Prashar Using certain rules and techniques it can be shown that: Directly following on from this statement, it was found that when N was odd, then G(N) was approximately 1 (the number of ways of partitioning a number as a sum of primes was 1). After further rigorous analysis, the key point of attention is that: N 2 log −3 N ≤ The number of ways N can be written as the sum of 3 primes This consequence of the equation suggests that when asymptotic analysis is used, a proof to Goldbach’s ternary conjecture can be achieved. The only problem is that the theory is only applicable to ‘sufficiently large’ numbers. It was later specified by one of Vinogradov’s students that greater than 314348907 was sufficiently large enough26. However, this is ridiculously large, as computerised methods have only checked numbers up to 1.2 × 10 18. Meaning that there is an extremely large range of numbers that have not been checked to see if they are the sum of three primes or not. Thus the ternary conjecture remains unproved. VIII) Conclusion 26 Goldbach's Weak Conjecture. June 24, 2009. http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture (accessed June 28, 2009). 21
• 22. IB Extended Essay: Maths Anil Prashar There are various ways to express a problem in mathematics. Through proper manipulation and development, a specific formula will allow for a proof. Some forms of the Goldbach conjecture have been lesser developed. There have been numerous methods of the sort. The methods explored in this essay have been either some of the simpler or more successful methods applied. The simple methods, such as probability, graphical methods and the smallest partition were useful ways of helping to look at ways in which conjectures may appear to be true. By examining that the probability of the Goldbach conjecture being true increased for larger values, the idea was enhanced that the theory was almost indefinitely true. However, there existed obvious flaws in the method. Probability can generally be summarised as a situation where something does or doesn’t happen (both can’t happen together). As the probability is never 1 for any of the partitions (although the larger values do tend to it) there is still a chance that there is one renegade value that would disprove Goldbach’s conjecture. Graphical methods proved useful in spotting links between certain parts of Goldbach’s conjecture, but the smallest prime number partition idea was of no use. The successful methods (i.e. the more rigorous methods) such as Schnirelmann’s theorem, Brun’s sieve, and Vinogradov’s theorem yielded much more beneficial results. Schnirelmann’s theorem proved that the conjecture was true in the respect that all numbers could indeed be split into the sum of a certain amount of prime numbers. In addition to this, Brun’s sieve and Vinogradov’s theorem helped to partially verify Goldbach’s ternary and binary conjectures, although the theorems were still only applicable to sufficiently large numbers. The magnitude of sufficiently large meant that the theory would have to be proved by computerised methods up to these values. A concept that mathematicians should be cautious of, as verifying a conjecture is one thing, but understanding why it is true is dependent on rigorous proof and analysis. IX) Bibliography 22
• 23. IB Extended Essay: Maths Anil Prashar Big O Notation. June 28, 2009. http://en.wikipedia.org/wiki/Big_O_notation (accessed July 1, 2009). Brun Sieve. May 19, 2009. http://en.wikipedia.org/wiki/Brun_sieve (accessed June 27, 2009). Caldwell, Chris K. Euclid's Proof of the Infinitude of Primes. http://primes.utm.edu/notes/proofs/infinite/euclids.html (accessed June 29, 2009). —. Goldbach's Conjecture. http://primes.utm.edu/glossary/xpage/GoldbachConjecture.html (accessed June 29, 2009). —. There are infinity many primes, but how big of an infinity? http://primes.utm.edu/infinity.shtml (accessed June 30, 2009). Charles, Denis Xavier. "Sieve Methods." 2000: 35. Eccles, Peter J. An Introduction to Mathematical Reasoning; numbers, sets and functions. Cambridge: Cambridge University Press, 2001. Goldbach's Weak Conjecture. June 24, 2009. http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture (accessed June 28, 2009). Herkommer, Mark. Goldbach Conjecture Research. May 24, 2004. http://www.petrospec-technologies.com/Herkommer/goldbach.htm (accessed June 30, 2009). Hofstadter, Douglas R. Goedel, Escher, Bach; an Eternal Golden Braid. London: Penguin Books Ltd., 2000. Inclusion-Exclusion Principle. June 26, 2009. http://en.wikipedia.org/wiki/Inclusion- exclusion_principle (accessed July 1, 2009). O'Bryant, Kevin. Mann's Theorem. http://mathworld.wolfram.com/MannsTheorem.html (accessed July 2, 2009). —. Schnirelmann's Theorem. http://mathworld.wolfram.com/SchnirelmannsTheorem.html (accessed July 2, 2009). PrimeFan. Chen's Theorem. http://planetmath.org/encyclopedia/ChensTheorem.html (accessed June 27, 2009). Schnirelmann's Theorem. June 27, 2009. http://en.wikipedia.org/wiki/Schnirelmann_constant#Schnirelmann.27s_theorem (accessed July 1, 2009). Shi, Kaida. "A New Method to Prove Goldbach Conjecture, Twin Primes Conjectures and Other Two Propositions." Zhejiang Ocean University, 2000: 2. T., Oliveira e Silva. Goldbach Conjecture Verification. July 14, 2008. http://www.ieeta.pt/~tos/goldbach.html (accessed July 1, 2009). 23
• 24. IB Extended Essay: Maths Anil Prashar Vinogradov's Theorem. June 24, 2009. http://en.wikipedia.org/wiki/Vinogradov %27s_theorem (accessed June 27, 2009). Von Mangoldt function. May 2, 2009. http://en.wikipedia.org/wiki/Von_Mangoldt_function (accessed June 30, 2009). Wang, Yuan. The Goldbach Conjecture. London: World Scientific Publishing Company; 2 Sub Edition, 2003. Weisstein, Eric W. Schnirelmann Constant. http://mathworld.wolfram.com/SchnirelmannConstant.html (accessed July 2, 2009). —. Schnirelmann Density. http://mathworld.wolfram.com/SchnirelmannDensity.html (accessed July 2, 2009). 24