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For example,
In 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture:
Euler, becoming interested in the problem, answered with a stronger version of the conjecture:
The former conjecture is known today as the 'weak' Goldbach conjecture, the latter as the 'strong' Goldbach conjecture. (The strong version implies the weak version, as any odd number greater than 5 can be obtained by adding 3 to any even number greater than 2). Without qualification, the strong version is meant. Both questions have remained unsolved ever since, although the weak form of the conjecture is much closer to resolution than the strong one.
The majority of mathematicians believe the conjecture (in both the weak and strong forms) to be true, at least for sufficiently large integers, mostly based on statistical considerations focusing on the probabilistic distribution of prime numbers: the bigger the number, the more ways there are available for that number to be represented as 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 chance of being prime. Thus if n 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 . This heuristic is non-rigorous for a number of reasons, for instance it assumes that the events that m and are prime are statistically independent of each other. Nevertheless, 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 this quantity goes to infinity as n increases, we expect that every large even integer has not just one representation as the sum of two primes, but in fact has very many such representations.
The above heuristic argument is actually somewhat inaccurate, because it ignores some correlations between the likelihood of m and being prime. For instance, if m is odd then is also odd, and odd numbers clearly are more likely to be prime than even numbers. Similarly, if n is divisible by 3, and m was already a prime distinct from 3, then 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, HardyGodfrey Harold Hardy ( February 7, 1877 December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. Non-mathematicians know him for two things: A Mathematician's Apology his essay from 19 and LittlewoodJohn Edensor Littlewood ( June 9 1885 September 6 1977) was a British mathematician. Littlewood was born in Rochester in Kent, and studied at Cambridge University. Most of his work was in the field of mathematical analysis. He collaborated for many years in 1923Centuries: 19th century 20th century 21st century Decades: 1870s 1880s 1890s 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s Years: 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 Events January 1 Grouping of all UK railway companies into four larg conjectured (as part of their famous Hardy-Littlewood prime tuple conjecture) that for any fixed c ≥ 2, the number of representations of a large integer n as the sum of c primes with should be asymptoticallyIn mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend''. It is sometimes expressed in the language of equivalence relations. For example, equal to
where the product is over all primes p, and is the number of solutions to the equation
in modular arithmeticModular arithmetic Group theory In mathematics, modular arithmetic is a system of arithmetic for certain equivalence classes of integers, called congruence classes . In modular arithmetic, numbers 'wrap around' after they reach a certain value (the modulu, subject to the constraints . This formula has been rigorously proven to be asymptotically valid for c ≥ 3 from the work of VinogradovIvan Matveevich Vinogradov ( September 14, 1891 March 20, 1983) was a Russian mathematician, who was one of the creators of modern analytic number theory, and also the dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district,, but is still only a conjecture when . In the latter case, the above formula simplifies to 0 when n is odd, and towhen n is even, where is the twin prime constant
This asymptotic is sometimes known as the extended Goldbach conjecture. The strong Goldbach conjecture is in fact very similar to the twin prime conjecture, and the two conjectures are believed to be of roughly comparable difficulty.