Goldbach's original conjecture, written in a 1742 letter to Euler, states that every Integer is
the Sum of three Primes. As re-expressed by Euler, an equivalent of this Conjecture (called
the ``strong'' Goldbach conjecture) asserts that all Positive Even Integers can be expressed
as the Sum of two Primes. Schnirelmann (1931) proved that every Even number can be written as the sum of not
more than 300,000 Primes (Dunham 1990), which seems a rather far cry from a proof for *two* Primes! The strong
Goldbach 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 sufficiently large 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 showed that all sufficiently large Even Numbers are the sum of a Prime and the Product of at most two 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 of
two Odd Primes, and every Integer the sum of exactly three distinct Primes. Let be the number
of representations of an Even Integer as the sum of two Primes. Then the ``extended'' Goldbach conjecture
states 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.

© 1996-9

1999-05-25