Goldbach Conjecture
Goldbach's original conjecture, written in a 1742 letter to Euler, 
states that every Integer 
isthe Euler, an equivalent of this Conjecture (calledthe ``strong'' Goldbach conjecture) asserts that all Positive Even Integers 
can be expressedas the Sum of two Primes. Schnirelmann (1931) proved that every Even number can be written as the sum of notmore than 300,000 Primes (Dunham 1990), which seems a rather far cry from a proof for two Primes! The strongGoldbach conjecture has been shown to be true up to 
by Sinisalo (1993).Pogorzelski (1977) claimed to have proven the Goldbach conjecture, but his proof is not generally accepted (Shanks 1993).
The conjecture that all Odd numbers 
are the Sum of three Odd Primes is called the``weak'' Goldbach conjecture. Vinogradov proved that all Odd Integers starting at some sufficientlylarge value are the Sum of three Primes (Guy 1994). The original ``sufficiently large'' 
was subsequently reduced to 
by Chen and Wang (1989). Chen (1973, 1978) also showedthat all sufficiently large Even Numbers are the sum of a Prime and the Product of at mosttwo Primes (Guy 1994, Courant and Robbins 1996).
It has been shown that if the weak Goldbach conjecture is false, then there are only a Finite number of exceptions.
Other variants of the Goldbach conjecture include the statements that every Even number 
is the Sum oftwo Odd Primes, and every Integer 
the sum of exactly three distinct Primes. Let 
be the numberof representations of an Even Integer 
as the sum of two Primes. Then the ``extended'' Goldbach conjecturestates that
where
is the Twin Primes Constant (Halberstam and Richert 1974).If the Goldbach conjecture is true, then for every number 
, there are Primes 
and 
such that
where
is the Totient Function (Guy 1994, p. 105).Vinogradov (1937ab, 1954) proved that every sufficiently large Odd Number is the sum of three Primes, and Estermann(1938) proves that almost all Even Numbers are the sums of two Primes.
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 64, 1987.
Chen, J.-R. ``On the Representation of a Large Even Number as the Sum of a Prime and the Product of at Most Two Primes.'' Sci. Sinica 16, 157-176, 1973.
Chen, J.-R. ``On the Representation of a Large Even Number as the Sum of a Prime and the Product of at Most Two Primes, II.'' Sci. Sinica 21, 421-430, 1978.
Chen, J.-R. and Wang, T.-Z. ``On the Goldbach Problem.'' Acta Math. Sinica 32, 702-718, 1989.
Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 30-31, 1996.
Devlin, K. Mathematics: The New Golden Age. London: Penguin Books, 1988.
Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 83, 1990.
Estermann, T. ``On Goldbach's Problem: Proof that Almost All Even Positive Integers are Sums of Two Primes.'' Proc. London Math. Soc. Ser. 2 44, 307-314, 1938.
Guy, R. K. ``Goldbach's Conjecture.'' §C1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 105-107, 1994.
Hardy, G. H. and Littlewood, J. E. ``Some Problems of Partitio Numerorum (V): A Further Contribution to the Study of Goldbach's Problem.'' Proc. London Math. Soc. Ser. 2 22, 46-56, 1924.
Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974.
Pogorzelski, H. A. ``Goldbach Conjecture.'' J. Reine Angew. Math. 292, 1-12, 1977.
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 30-31 and 222, 1985.
Sinisalo, M. K. ``Checking the Goldbach Conjecture up to 
.'' Math. Comput. 61, 931-934, 1993.
Vinogradov, I. M. ``Representation of an Odd Number as a Sum of Three Primes.'' Comtes rendus (Doklady) de l'Académie des Sciences de l'U.R.S.S. 15, 169-172, 1937a.
Vinogradov, I. ``Some Theorems Concerning the Theory of Primes.'' Recueil Math. 2, 179-195, 1937b.
Vinogradov, I. M. The Method of Trigonometrical Sums in the Theory of Numbers. London: Interscience, p. 67, 1954.
Yuan, W. Goldbach Conjecture. Singapore: World Scientific, 1984.
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