Goldbach's conjecture
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In mathematics, Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:
- Every even number greater than 2 can be written as the sum of two primes. (The same prime may be used twice.)
For example,
- 4 = 2 + 2
- 6 = 3 + 3
- 8 = 3 + 5
- 10 = 3 + 7 = 5 + 5
- 12 = 5 + 7
- 14 = 3 + 11 = 7 + 7
- etc.
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Origins
In 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture:
- Every integer greater than 2 can be written as the sum of three primes.
He considered 1 to be a prime number, a convention subsequently abandoned. So today, Goldbach's original conjecture would be written:
- Every integer greater than 5 can be written as the sum of three primes.
Euler, becoming interested in the problem, answered with an equivalent version of the conjecture:
- Every even number greater than 2 can be written as the sum of two primes.
The former conjecture is today known as the "ternary" Goldbach conjecture, the latter as the "strong" Goldbach conjecture. The conjecture that all odd numbers greater than 9 are the sum of three odd primes is called the "weak" Goldbach conjecture. Both questions have remained unsolved ever since, although the weak form of the conjecture is much closer to resolution than the strong one.
Heuristic justification
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 <math>1/\ln m<math> 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 <math>{1 \over \ln m \ln (n-m)}<math>. This heuristic is non-rigorous for a number of reasons, for instance it assumes that the events that m and <math>n-m<math> 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
- <math> \sum_{m=3}^{n/2} \frac{1}{\ln m} {1 \over \ln (n-m)} \approx \frac{n}{2 \ln^2 n}.<math>
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 <math>n-m<math> being prime. For instance, if m is odd then <math>n-m<math> 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 <math>n-m<math> 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, Hardy and Littlewood in 1923 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 <math>n=p_1+\dotsb+p_c<math> with <math>p_1 \leq \dotsb \leq p_c<math> should be asymptotically equal to
- <math> \left(\prod_p \frac{p \gamma_{c,p}(n)}{(p-1)^c}\right)
\int_{2 \leq x_1 \leq \dotsb \leq x_c: x_1+\ldots+x_c = n} \frac{dx_1 \ldots dx_{c-1}}{\ln x_1 \ldots \ln x_c}<math>
where the product is over all primes p, and <math>\gamma_{c,p}(n)<math> is the number of solutions to the equation <math>n = q_1 + \ldots + q_c \mod p<math> in modular arithmetic, subject to the constraints <math>q_1,\ldots,q_c \neq 0 \mod p<math>. This formula has been rigorously proven to be asymptotically valid for c ≥ 3 from the work of Vinogradov, but is still only a conjecture when <math>c=2<math>. In the latter case, the above formula simplifies to 0 when n is odd, and to
- <math> 2 \Pi_2 \left(\prod_{p|n; p \geq 3} \frac{p-1}{p-2}\right) \int_2^n \frac{dx}{\ln^2 x}
\approx 2 \Pi_2 \left(\prod_{p|n; p \geq 3} \frac{p-1}{p-2}\right) \frac{n}{\ln^2 n} <math>
when n is even, where <math>\Pi_2<math> is the twin prime constant
- <math> \Pi_2 := \prod_{p \geq 3} \left(1 - \frac{1}{(p-1)^2}\right) = 0.660161858\ldots.<math>
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.
Rigorous results
For small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, N. Pipping in 1938 laboriously verified the conjecture up to <math>n \leq 10^5<math>. With the advent of computers, many more small values of n have been checked; T. Oliveira e Silva is running a distributed computer search that has verified the conjecture up to <math>n \leq 2 \times 10^{17}<math> (as of March 2004).
The weak Goldbach conjecture is fairly close to resolution. In 1923, Hardy and Littlewood showed that under the assumption of the generalized Riemann hypothesis (GRH), every sufficiently large odd number was the sum of three primes. In 1937, Ivan Vinogradov removed the hypothesis of GRH and proved that every sufficiently large odd number n is the sum of three primes. Vinogradov's student, K. Borodzin, quantified the phrase sufficiently large, showing that <math>n>3^{3^{15}}<math> would suffice. This bound has since been lowered a number of times, with the currently best known result due to Chen and Wang in 1989, who proved that every odd number <math>n>e^{e^{11.503}}\approx 3.33 \times 10^{43000}<math> is the sum of three primes. In principle, this leaves only a finite number of cases to check, but this is far too large a number to be handled by computer search (which, as mentioned earlier, has only reached as far as <math>2 \times 10^{17}<math> for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture). In 1997 Deshoulliers, Effinger, Te Riele, and Zinoviev were able to close the gap and prove that all odd numbers (greater than 5) are the sum of three primes, but only by assuming GRH again.
The strong Goldbach conjecture is much more difficult. The work of Vinogradov in 1937 and Theodor Estermann (1902-1991) in 1938 showed that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers which can be so written tends towards 1). In 1939, L.G. Schnirelmann proved that every even number n ≥ 4 can be written as the sum of at most 300,000 primes. This result was subsequently improved by many authors; currently, the best known result is due to Ramaré, who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most six primes.
Chen Jingrun showed in 1966 using the methods of sieve theory that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes)—e.g., 100 = 23 + 7·11.
In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers were expressible as the sum of two primes. More precisely, they showed that there existed positive constant <math>c, C<math> such that for all sufficiently large numbers N, every even number less than N is the sum of two primes, with at most <math>C N^{1-c}<math> exceptions. In particular, the set of even integers which are not the sum of two primes has density zero.
Trivia
Doug Lenat's Automated Mathematician rediscovered Goldbach's Conjecture in 1982. This is considered one of the earliest demonstrations that artificial intelligences are capable of scientific discovery (but see the discussion at Automated Mathematician).
In order to generate publicity for the book Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis, British publisher Tony Faber offered a $1,000,000 prize for a proof of the conjecture in 2000. The prize was only to be paid for proofs submitted for publication before April 2002. The prize was never claimed.
One can pose similar questions when primes are replaced by other special sets of numbers, such as the squares. For instance, it was proven by Lagrange that every positive integer is the sum of four squares. See Waring's problem.
Attempted proofs
As with many famous conjectures in mathematics, there are a number of purported proofs of the Goldbach conjecture, none which are currently accepted by mainstream mathematicians.
Because it is easily understood by laymen, Goldbach's conjecture is a popular target for pseudomathematicians who attempt to prove it, sometimes even disprove it, using only high-school-level mathematics. It shares this fate with the four-color theorem and Fermat's last theorem, each of which also has an easily stated problem, but a current proof which is extraordinarily elaborate.
It is possible that problems like Goldbach's conjecture may yield to simple methods, but given the amount of professional attention given to these problems, it is unlikely that the first solution will be easy to find.
External links
- Goldbach's conjecture (http://www.utm.edu/research/primes/glossary/GoldbachConjecture.html), part of Chris Caldwell's Prime Pages (http://primes.utm.edu/).
- A million-dollar maths question (http://www.times-archive.co.uk/news/pages/tim/2000/03/16/timfeafea02004.html). Article by Anjana Ahuja in The Times, March 16, 2000.
- Help verify the Goldbach conjecture (http://www.ieeta.pt/~tos/goldbach/help.html), Tomás Oliveira e Silva's distributed computer search.
- Online tool (http://wims.unice.fr/wims/wims.cgi?module=tool/number/goldbach.en) to test Goldbach's conjecture on submitted integers.
References
- Apostolos Doxiadis: Uncle Petros and Goldbach's Conjecture. ISBN 1582341281.
- J.-M. Deshouillers; G. Effinger; H. te Riele; D. Zinoviev, A complete Vinogradov 3-primes theorem under the Riemann hypothesis, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 99--104 (electronic).de:Goldbachsche Vermutung
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