Could G. H. Hardy make a product of two primes so big he couldn't find out which? This question of course began as a slightly irreverent play on the riddle, "Can God make a stone so big He can't lift it?" Then I wondered about the answer.

Is it possible to exhibit a number that is provably a product of exactly two primes without knowing which primes? More to the point, without having the theoretical and technological means to compute which primes? If so, how is that done, and how old are the results that are needed in order to do it? And if not, does recent theory and technology make it impossible?
The magic phrase for this problem is 'Zero-Knowledge Proof', and a bit of hunting around suggests that there is in fact a zero-knowledge scheme for the product of two primes - not an exact proof, but a statistical one (so that it can be verified to arbitrarily high probability even with an adversarial prover).  From the excerpt for the paper An efficient non-interactive statistical zero-knowledge proof system for quasi-safe prime products by Gennaro, Micciancio and Rabin:


  [...] we present the first simple and efficient zero-knowledge proof that an alleged RSA modulus is of the correct form, i.e. the product of two primes.


It looks like the proof goes in stages; first, a zero-knowledge proof that a number $n$ is squarefree by exhibiting the ability to extract $n$th roots of numbers (numbers $n$ that aren't square-free have $\mathrm{GCD}(n,\phi(n))&gt;1$ and thus only a small fraction of numbers will have $n$th roots mod $n$), then a zero-knowledge proof that a number is a product of two prime powers (both of which are inequal to each other and inequal to 1 mod 8), where the verifier supplies a number $x$ and the prover provides a square root for one of $\pm x, \pm 2x$; by quadratic reciprocity, this is always possible for numbers satisfying the condition, whereas products $m$ of three or more prime factors have at most one-eighth of the numbers less than $m$ as quadratic residues, so a randomly-chosen $x$ will fail to have any of $\{\pm x, \pm 2x\}$ be a square at least half the time.  Combining the two zero-knowledge proofs gives a zero-knowledge proof for (a subset of) the property 'product of at most two primes', and then combining this with the standard primality tests gives a proof for (a subset of) 'product of exactly two primes'.

Note that this is a statistical proof, though, not an exact one; you can be convinced of the result to an arbitrary degree of certainty but it's not a proof in the classical sense.  For more details on zero-knowledge proofs in general, the Wikipedia page is a decent starting point.
Depends on what you mean by 'exhibit a number'. How about $14$ iff for all $N&lt{10^{10^{99999}}}$, $2N$ is a sum of two primes, else $15$?

This is a well-defined, computable and unique number.

In what sense is the string $56566633746456456$ a better representation of a number then "$x=14$ iff Goldbach is true for all $N&lt{10^{10^{99999}}}$, else $x=15$", other then the artificial human base-10 convention ?