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43-407: X and Y are two whole numbers greater than 1, and Y > X . Their sum is not greater than 100. S and P are two mathematicians (and consequently perfect logicians); S knows the sum X + Y and P knows the product X × Y . Both S and P know all the information in this paragraph. In the following conversation, both participants are always telling the truth: What are X and Y ? The problem

86-460: A 2-split. There is no need for any advanced knowledge like Goldbach's conjecture or the fact that for the product B·C of such a 2-split to be unique (i.e. there are no other two numbers that also when multiplied yield the same result). But with Goldbach's conjecture, along with the fact that P would immediately know X and Y if their product were a semiprime , it can be deduced that the sum x+y cannot be even, since every even number can be written as

129-467: A distributed computer search that has verified the conjecture for n ≤ 4 × 10 (and double-checked up to 4 × 10 ) as of 2013. One record from this search is that 3 325 581 707 333 960 528 is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781. Cully-Hugill and Dudek prove a (partial and conditional) result on the Riemann hypothesis: there exists

172-437: A given sum A . When there are many solutions, that is, for higher t , some solutions coincide with those for the original problem with Y > X > 1, for example X = 16, Y = 163. If the condition X + Y ≤ t for some threshold t is exchanged for X·Y ≤ u instead, the problem changes appearance. It becomes easier to solve with less calculations required. A reasonable value for u could be u = t · t /4 for

215-473: A large integer n as the sum of c primes n = p 1 + ⋯ + p c with p 1 ≤ ⋯ ≤ p c should be asymptotically equal to where the product is over all primes p , and γ c , p ( n ) is the number of solutions to the equation n = q 1 + ⋯ + q c mod p in modular arithmetic , subject to the constraints q 1 , …, q c ≠ 0 mod p . This formula has been rigorously proven to be asymptotically valid for c ≥ 3 from

258-437: A prime, under which 1 is excluded. A modern version of the first conjecture is: A modern version of the marginal conjecture is: And a modern version of Goldbach's older conjecture of which Euler reminded him is: These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integer N = p + 1 larger than 4, for p a prime, that could not be expressed as

301-464: A specific counterexample N ). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement. The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the " strong ", "even", or "binary" Goldbach conjecture. A weaker form of

344-476: A sum of two odd primes in the interval (x, x + 9696 log^2 x] for all x ≥ 2. Goldbach's Conjecture ( Chinese : 哥德巴赫猜想 ) is the title of the biography of Chinese mathematician and number theorist Chen Jingrun , written by Xu Chi . The conjecture is a central point in the plot of the 1992 novel Uncle Petros and Goldbach's Conjecture by Greek author Apostolos Doxiadis , in the short story " Sixty Million Trillion Combinations " by Isaac Asimov and also in

387-502: Is a 27 state Turing machine that halts if and only if Goldbach's conjecture is false. Hence if BB(27) was known, and the Turing machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(27) will be very hard to compute, at least as difficult as settling

430-554: Is a large even integer and m is a number between 3 and ⁠ n / 2 ⁠ , then one might expect the probability of m and n − m simultaneously being prime to be ⁠ 1 / ln m ln( n − m ) ⁠ . If one pursues this heuristic, one might expect the total number of ways to write a large even integer n as the sum of two odd primes to be roughly Since ln n ≪ √ n , this quantity goes to infinity as n increases, and one would expect that every large even integer has not just one representation as

473-474: Is a sum of three primes. However, the converse implication and thus the strong Goldbach conjecture would remain unproven if Helfgott's proof is correct. For small values of n , the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, in 1938, Nils Pipping laboriously verified the conjecture up to n = 100 000 . With the advent of computers, many more values of n have been checked; T. Oliveira e Silva ran

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516-463: Is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it. The strong Goldbach conjecture is much more difficult than the weak Goldbach conjecture , which says that every integer (equivalently, every odd integer) greater than 5 is the sum of three primes. Using Vinogradov's method , Nikolai Chudakov , Johannes van der Corput , and Theodor Estermann showed (1937–1938) that almost all even numbers can be written as

559-414: Is altered, the number of solutions may change. For X + Y < 62, there is no solution. This might seem counter-intuitive at first since the solution X = 4, Y = 13 fits within the boundary. But by the exclusion of products with factors that sum to numbers between these boundaries, there are no longer multiple ways of factoring all non-solutions, leading to the information yielding no solution at all to

602-427: Is changed to Y > X > 2, there is a unique solution for thresholds X + Y ≤ t for 124 < t < 5045, after which there are multiple solutions. At 124 and below, there are no solutions. It is not surprising that the threshold for a solution has gone up. Intuitively, the problem space got "sparser" when the prime number 2 is no longer available as the factor X , creating fewer possible products X·Y from

645-411: Is divisible by 3, and m was already a prime other than 3, then n − m would also be coprime to 3 and thus be slightly more likely to be prime than a general number. Pursuing this type of analysis more carefully, G. H. Hardy and John Edensor Littlewood in 1923 conjectured (as part of their Hardy–Littlewood prime tuple conjecture ) that for any fixed c ≥ 2 , the number of representations of

688-518: Is one of the oldest and best-known unsolved problems in number theory and all of mathematics . It states that every even natural number greater than 2 is the sum of two prime numbers . The conjecture has been shown to hold for all integers less than 4 × 10 but remains unproven despite considerable effort. On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed

731-421: Is rather easily solved once the concepts and perspectives are made clear. There are three parties involved, S, P, and O. S knows the sum X+Y , P knows the product X·Y , and the observer O knows nothing more than the original problem statement. All three parties keep the same information but interpret it differently. Then it becomes a game of information. Let us call the split of a number A into two terms A=B+C

774-546: Is the sum of two primes, with at most CN exceptions. In particular, the set of even integers that are not the sum of two primes has density zero. In 1951, Yuri Linnik proved the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. János Pintz and Imre Ruzsa found in 2020 that K = 8 works. Assuming the generalized Riemann hypothesis , K = 7 also works, as shown by Roger Heath-Brown and Jan-Christoph Schlage-Puchta in 2002. A proof for

817-413: The 2008 mystery novel No One You Know by Michelle Richmond . Goldbach's conjecture is part of the plot of the 2007 Spanish film Fermat's Room . Goldbach's conjecture is featured as the main topic of research of actress Ella Rumpf 's character Marguerite in the 2023 French-Swiss film Marguerite's Theorem . Each of the three conjectures has a natural analog in terms of the modern definition of

860-702: The Goldbach conjecture. List of unsolved problems in mathematics Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics , such as theoretical physics , computer science , algebra , analysis , combinatorics , algebraic , differential , discrete and Euclidean geometries , graph theory , group theory , model theory , number theory , set theory , Ramsey theory , dynamical systems , and partial differential equations . Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for

903-475: The assumption of the generalized Riemann hypothesis that the number of even numbers up to X violating the Goldbach conjecture is much less than X for small c . In 1948, using sieve theory methods, Alfréd Rényi showed that every sufficiently large even number can be written as the sum of a prime and an almost prime with at most K factors. Chen Jingrun showed in 1973 using sieve theory that every sufficiently large even number can be written as

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946-494: The corresponding t based on the largest product of two factors whose sum are t being ( t /2)·( t /2). Now the problem has a unique solution in the ranges 47 < t < 60, 71 < t < 80, 107 < t < 128, and 131 < t < 144 and no solution below that threshold. The results for the alternative formulation do not coincide with those of the original formulation, neither in number of solutions, nor in content. Goldbach%27s conjecture Goldbach's conjecture

989-408: The following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number , so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: ... eine jede Zahl, die grösser ist als 2, ein aggregatum trium numerorum primorum sey. Every integer greater than 2 can be written as

1032-536: The lists have been associated with prizes for the discoverers of solutions. Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date: The seventh problem, the Poincaré conjecture , was solved by Grigori Perelman in 2003. However, a generalization called the smooth four-dimensional Poincaré conjecture —that is, whether

1075-515: The numbers requiring the largest number of primes in their greedy representations. Similar problems to Goldbach's conjecture exist in which primes are replaced by other particular sets of numbers, such as the squares: Goldbach's conjecture is used when studying computation complexity. The connection is made through the Busy Beaver function, where BB( n ) is the maximum number of steps taken by any n state Turing machine that halts. There

1118-415: The only possible solution based on the fact that S was able to determine it. Note that for instance, if S had been told 97 (48 + 49) and P was told 2352 (48 * 49), P would be able to deduce the only possible solution, but S would not, as 44 & 53 would still be a logically possible alternative. The problem can be generalized. The bound X + Y ≤ 100 is chosen rather deliberately. If the limit of X + Y

1161-416: The problem is not solvable in the sense that there is no longer a unique solution. Similarly, if X + Y ≤ 1970 or higher a third solution appears ( X = 16, Y = 73). All of these three solutions contain one prime number. The first solution with no prime number is the fourth which appears at X + Y ≤ 2522 or higher with values X = 16 = 2·2·2·2 and Y = 111 = 3·37. If the condition Y > X > 1

1204-453: The problem. For example, if X = 2, Y = 62, X + Y = 64, X · Y =124 is not considered, then there remains only one product of 124, viz. 4·31, yielding a sum of 35. Then 35 is eliminated when S declares that P cannot know the factors of the product, which it would not have been if the sum of 64 was allowed. On the other hand, when the limit is X + Y ≤ 1685 or higher, there appears a second solution X = 4, Y = 61. Thus, from then on,

1247-434: The product, P can rule out 2 x 26, as in that case the sum is 28, and even sums are ruled out by Goldbach's conjecture as above. So P now knows the numbers are 4 and 13 and tells S that he knows the numbers. From this, S now knows that of the possible pairs based on the sum (viz. 2+15, 3+14, 4+13, 5+12, 6+11, 7+10, 8+9) only one has a product that would allow P to deduce the answer, that being 4 + 13. The reader can then deduce

1290-466: The second modern statement, known as " Goldbach's weak conjecture ", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that Statistical considerations that focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture (in both the weak and strong forms) for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as

1333-589: The solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems , receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance. Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases,

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1376-419: The sum of either two primes, or a prime and a semiprime (the product of two primes). See Chen's theorem for further information. In 1975, Hugh Lowell Montgomery and Bob Vaughan showed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constants c and C such that for all sufficiently large numbers N , every even number less than N

1419-620: The sum of three primes. Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (" ... so Ew vormals mit mir communicirt haben ... "), in which Goldbach had remarked that the first of those two conjectures would follow from the statement This is in fact equivalent to his second, marginal conjecture. In the letter dated 30 June 1742, Euler stated: Dass ... ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann. That ... every even integer

1462-427: The sum of two or three other numbers, and the more "likely" it becomes that at least one of these representations consists entirely of primes. A very crude version of the heuristic probabilistic argument (for the strong form of the Goldbach conjecture) is as follows. The prime number theorem asserts that an integer m selected at random has roughly a ⁠ 1 / ln m ⁠ chance of being prime. Thus if n

1505-518: The sum of two prime numbers. The product of those two numbers would then be a semiprime. The following steps give the solution: The solution has X and Y as 4 and 13, with P initially knowing the product is 52 and S knowing the sum is 17. Initially P does not know the solution, since and S knows that P does not know the solution since all the possible sums to 17 within the constraints produce similarly ambiguous products. However once P knows that S believes there are multiple possible solutions given

1548-494: The sum of two primes (in the sense that the fraction of even numbers up to some N which can be so written tends towards 1 as N increases). In 1930, Lev Schnirelmann proved that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant; see Schnirelmann density . Schnirelmann's constant is the lowest number C with this property. Schnirelmann himself obtained C < 800 000 . This result

1591-403: The sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer n , could not possibly rule out the existence of such

1634-412: The sum of two primes, and also that the number of these representations depend strongly on the value modulo 3 of the number. Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime at each step. The Pillai sequence tracks

1677-448: The sum of two primes, but in fact very many such representations. This heuristic argument is actually somewhat inaccurate because it assumes that the events of m and n − m being prime are statistically independent of each other. For instance, if m is odd, then n − m is also odd, and if m is even, then n − m is even, a non-trivial relation because, besides the number 2, only odd numbers can be prime. Similarly, if n

1720-414: The two conjectures are believed to be of roughly comparable difficulty. The Goldbach partition function is the function that associates to each even integer the number of ways it can be decomposed into a sum of two primes. Its graph looks like a comet and is therefore called Goldbach's comet . Goldbach's comet suggests tight upper and lower bounds on the number of representations of an even number as

1763-427: The weak conjecture was submitted in 2013 by Harald Helfgott to Annals of Mathematics Studies series. Although the article was accepted, Helfgott decided to undertake the major modifications suggested by the referee. Despite several revisions, Helfgott's proof has not yet appeared in a peer-reviewed publication. The weak conjecture is implied by the strong conjecture, as if n − 3 is a sum of two primes, then n

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1806-402: The work of Ivan Matveevich Vinogradov , but is still only a conjecture when c = 2 . In the latter case, the above formula simplifies to 0 when n is odd, and to when n is even, where Π 2 is Hardy–Littlewood's twin prime constant This is sometimes known as the extended Goldbach conjecture . The strong Goldbach conjecture is in fact very similar to the twin prime conjecture, and

1849-403: Was subsequently enhanced by many authors, such as Olivier Ramaré , who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most 6 primes. The best known result currently stems from the proof of the weak Goldbach conjecture by Harald Helfgott , which directly implies that every even number n ≥ 4 is the sum of at most 4 primes. In 1924, Hardy and Littlewood showed under

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