that point, the name "Goldberg Variations" will start to shift slightly in
meaning, to include not only the known ones, but also any others
which might eventually turn up. Their number-call it 'g'-is certain
to be finite, wouldn't you agree?-but merely knowing that g is finite
isn't the same as knowing how big g is. Consequently, this information
won't tell us when the last Goldberg Variation has been located.
Tortoise: That is certainly true.
Achilles: Tell me-when was it that Bach wrote these celebrated varia-
tions?
Tortoise: It all happened in the year 1742, when he was Cantor in Leipzig.
Achilles: 1742? Hmm ... That number rings a bell.
Tortoise: It ought to, for it happens to be a rather interesting number,
being a sum of two odd primes: 1729 and 13.
Achilles: By thunder! What a curious fact! I wonder how often one runs
across an even number with that property. Let's see ...
6 = 3 + 3
8 = 3 + 5
10= 3+7= 5+5
12 = 5 + 7
14 = 3 + 11 = 7 + 7
16 = 3 + 13 = 5 + 11
18 = 5 + 13 = 7 + 11
20 = 3 + 17 = 7 + 13
22 = 3 + 19 = 5 + 17 = 11 + 11
24 = 5 + 19 = 7 + 17 = 11 + 13
26 = 3 + 23 = 7 + 19 = 13 + 13
28 = 5 + 23 = 11 + 17
30 = 7 + 23 = 11 + 19 = 13 + 17
Now what do you know-according to my little table here, it seems to
be quite a common occurrence. Yet I don't discern any simple regular-
ity in the table so far.
Tortoise: Perhaps there is no regularity to be discerned.
Achilles: But of course there is! I am just not clever enough to spot it right
off the bat.
Tortoise: You seem quite convinced of it.
Achilles: There's no doubt in my mind. I wonder ... Could it be that ALL
even numbers (except 4) can be written as a sum of two odd primes?
Tortoise: Hmm ... That question rings a bell ... Ah, I know why! You're
not the first person to ask that question. Why, as a matter of fact, in the
year 1742, a mathematical amateur put forth this very question in a-
Achilles: Did you say 1742? Excuse me for interrupting, but I just noticed
that 1742 happens to be a rather interesting number, being a differ-
ence of two odd primes: 1747 and 5.
Tortoise: By thunder! What a curious fact! I wonder how often one runs
across an even number with that property.
Aria with Diverse Variations^393