Achilles: But please don't let me dIstract you from your story.
Tortoise: Oh, yes-as I was saying, in 1742, a certain mathematical
amateur, whose name escapes me momentarily, sent a letter to Euler,
who at the time was at the court of King Frederick the Great in
Potsdam, and-well, shall I tell you the story? It is not without charm.
Achilles: In that case, by all means, do!
Tortoise: Very well. In his letter, this dabbler in number theory pro-
pounded an unproved conjecture to the great Euler: "Every even
number can be represented as a sum of two odd primes." Now what
was that fellow's name?
Achilles: I vaguely recollect the story, from some number theory book or
other. Wasn't the fellow named ·'Kupfergodel"?
Tortoise: Hmm ... No, that sounds too long.
Achilles: Could it have been "Silberescher"?
Tortoise: No, that's not it, either. There's a name on the tip of my
tongue-ah-ah-oh yes! It was "Goldbach"! Goldbach was the fellow.
Achilles: I knew it was something like that.
Tortoise: Yes-your guesses helped jog my memory. It's quite odd, how
one occasionally has to hunt around in one's memory as if for a book in
a library without call numbers .. But let us get back to 1742.
Achilles: Indeed, let's. I wanted to ask you: did Euler ever prove that this
guess by Goldbach was right?
Tortoise: Curiously enough, he never even considered it worthwhile work-
ing on. However, his disdain was not shared by all mathematicians. In
fact, it caught the fancy of many, and became known as the "Goldbach
Conjecture".
Achilles: Has it ever been proven correct?
Tortoise: No, it hasn't. But there have been some remarkable near misses.
For instance, in 1931 the Russian number theorist Schnirelmann
proved that any number-even or odd-can be represented as the sum
of not more than 300,000 primes.
Achilles: What a strange result. Of what good is it?
Tortoise: It has brought the problem into the domain of the finite. Previ-
ous to Schnirelmann's proof, it was conceivable that as you took larger
and larger even numbers, they would require more and more primes
to represent them. Some even number might take a trillion primes to
represent it! Now it is known that that is not so-a sum of 300,000
primes (or fewer) will always suffice.
Achilles: I see.
Tortoise: Then in 1937, a sly fellow named Vinogradov-a Russian
too-managed to establish something far closer to the desired result:
namely, every sufficiently large ODD number can be represented as
a sum of no more than THREE odd primes. For example,
1937 = 641 + 643 + 653. We could say that an odd number which is
representable as a sum of three odd primes has "the Vinogradov
property". Thus, all sufficiently large odd numbers have the Vino-
gradov property.
(^394) Aria with Diverse Variations