Goldbach's Conjecture Website Links For
Conjecture 		 

Information About

Goldbach's Conjecture
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



:Every Even Integer 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.


ORIGINS


In 1742 , the Prussian Mathematician Christian Goldbach wrote a letter to Leonhard Euler {Link without Title} 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 1/\ln m\,\! 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 1 \big / \big m \,\ln (n-m)\big . This heuristic is non-rigorous for a number of reasons, for instance it assumes that the events that ''m'' and n-m 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

: \sum_{m=3}^{n/2} rac{1}{\ln m} {1 \over \ln (n-m)} \approx rac{n}{2 \ln^2 n}.

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 n-m being prime. 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 2) only odd numbers can be prime. Similarly, if ''n'' is divisible by 3, and ''m'' was already a prime distinct from 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, 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 n=p_1+\dotsb+p_c with p_1 \leq \dotsb \leq p_c should be Asymptotically equal to

: \left(\prod_p rac{p \gamma_{c,p}(n)}{(p-1)^c} ight)
\int_{2 \leq x_1 \leq \dotsb \leq x_c: x_1+\ldots+x_c = n} rac{dx_1 \ldots dx_{c-1}}{\ln x_1 \ldots \ln x_c}

where the product is over all primes ''p'', and \gamma_{c,p}(n) is the number of solutions to the equation
n = q_1 + \ldots + q_c \mod p in Modular Arithmetic , subject to the Constraints q_1,\ldots,q_c
eq 0 \mod p. 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 c=2. In the latter case, the above formula simplifies to 0 when ''n'' is odd, and to



n>e^{3100}\approx 2 imes 10^{1346} 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 3 imes 10^{17} 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 1930 , 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 Olivier 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 1973 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 constants c, C such that for all sufficiently large numbers ''N'', every even number less than ''N'' is the sum of two primes, with at most C N^{1-c} exceptions. In particular, the set of even integers which are not the sum of two primes has density zero.

Roger Heath-Brown and Jan-Christoph Schlag-Puchta showed in 2002, that every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2.


TRIVIA


Doug Lenat 's Automated Mathematician rediscovered Goldbach's Conjecture in 1982. This is considered one of the earliest demonstrations that Artificial Intelligence s 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 .

The television drama '' Lewis '' featured a mathematics professor at Oxford University who had won the Fields Medal for his work on Goldbach's conjecture, which was a main plot feature.

SF author Stephen Baxter penned a short story entitled "Planck Zero" in which the Goldbach Conjecture features prominently. An Artificial Intelligence that can do an infinite number of calculations per second is asked to solve the Goldbach Conjecture by brute force. That is, merely examining ALL integers, and finding any cases where they don't meet the criteria of the conjecture. This story appeared in the collection Vacuum Diagrams.


ATTEMPTED PROOFS


As with many famous conjectures in mathematics, there are a number of purported proofs of the Goldbach conjecture, none of 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. Moreover it is thought that a simple proof of these two theorems is simply unattainable.

It is possible Goldbach's conjecture may yield to simple methods, but given the amount of professional attention paid to the conjecture, it is unlikely that a proof (or, alternatively, a counter-example) will be easy to find.


EXTERNAL LINKS




SEE ALSO




REFERENCES