A famous conjecture.

There are many unproved conjectures regarding prime numbers. One of these was made by Christian Goldbach (1690-1764) in 1742 in a letter to the great Swiss mathematician  Euler.

Goldbach had observed that every even integer, except 2, seemed representable as the sum of two primes. Thus 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3,..., 48 = 29 + 19,..., 100 = 97 + 3, and so forth.
Though Euler brought his remarkable powers to bear upon this problem, he was unable to resolve the conjecture one way or the other. In fact, to this day the problem remains intractable, though some progress on it has recently been made.
In 1931 the Russian mathematician L. Schnirelmann (1905-1938)  showed   that every positive integer, prime or composite,   can be represented as the sum of not more than
300,000 primes! Somewhat later, the Russian mathematician I. M. Vinogradoff (contemporary) showed that there exists a positive integer N such that any integer n > N can be expressed as the sum of at most four primes, but his proof in no way permits us to appraise the size of N.

Goldbach seems to have been an industrious correspondent and to have had the respect of many of the top mathematicians of his day. It was in a letter to Goldbach in 1746, for example, that Euler first announced his remarkable discovery that ii (where i= I/ - 1) is a real number. But today, about the only mention of Goldbach in the history of mathematics concerns his teasing conjecture given above.