Achilles: I think you should call it the "Achilles property". After all, I
suggested the problem.
Tortoise: I was just about to propose that we should say that a number
which LACKS the Tortoise property has the "Achilles property".
Achilles: Well, all right ...
Tortoise: Now consider, for instance, whether 1 trillion has the Goldbach
property or the Tortoise property. Of course, it may have both.
Achilles: I can consider it, but I doubt whether I can give you an answer to
either question.
Tortoise: Don't give up so soon. Suppose I asked you to answer one or the
other question. Which one would you pick to work on?
Achilles: I suppose I would flip a coin. I don't see much difference be-
tween them.
Tortoise: Aha! But there's a world of difference! If you pick the Goldbach
property, involving SUMS of primes, then you are limited to using
primes which are bounded between 2 and 1 trillion, right?
Achilles: Of course.
Tortoise: So your search for a representation for 1 trillion as a sum of two
primes is GUARANTEED TO TERMINATE.
Achilles: Ahhh! I see your point. Whereas if I chose to work on represent-
ing 1 trillion as the DIFFERENCE of two primes, I would not have any
bound on the size of the primes involved. They might be so big that it
would take me a trillion years to find them.
Tortoise: Or then again, they might not even EXIST! After all, that's what
the question was asking-do such primes exist? It wasn't of much
concern how big they might tum out to be.
Achilles: You're right. If they didn't exist, then a search process would
lead on forever, ne\'er answering yes, and never answering no. And
nevertheless, the answer would be no.
Tortoise: So if you have some number, and you wish to test whether it has
the Goldbach property or the Tortoise property, the difference be-
tween the two tests will be this: in the former, the search involved is
GUARANTEED TO TERMINATE; in the latter, it is POTENTIALLY
ENDLESS-there are no guarantees of any type. It might just go merrily
on forever, without yielding an answer. And yet, on the other hand, in
some cases, it might stop on the first step.
Achilles: I see there is a rather vast difference between the Goldbach and
Tortoise properties.
Tortoise: Yes, the two similar problems concern these vastly different
properties. The Goldbach Conjecture is to the effect that all even
numbers have the Goldbach property; the Goldbach Variation
suggests that all even numbers have the Tortoise property. Both prob-
lems are unsolved, but what is interesting is that although they sound
very much alike, they involve properties of whole numbers which are
quite different.
Achilles: I see what you mean. The Goldbach property is a detectable, or
(^396) Aria with Diverse Variations