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Here's another fun argument. The question boils down to "how common are primes?" And the answer is, very common. We can define a subset of positive integers as "small" if the sum of their reciprocals converges, and "large" otherwise. For example, the set of all positive integers is large (because the harmonic series diverges), and the complement of any small set is large. Well, it's possible to prove that the set of all primes is large, while the set of all numbers not containing some digit (say, 7) is small. So once you go far enough from zero, there are way more primes than there are numbers not containing 7. Now it doesn't sound as surprising that you can make squares out of primes, does it?
Say, that we have N-1 lines, with N-1 primes. Each N digits. What we now need is an N digit prime number to put it below.
Its most significant digit may be 1, 3, 7 or 9. Otherwise, the leftmost vertical number wouldn't be prime. If the sum of all N-1 other rightmost digits is X, then:
If X mod 3 = 0, then just 1 and 7 are possible, otherwise the leftmost vertical would be divisible by 3. If X mod 3 = 1, then 1, 3, 7 and 9 are possible. If X mod 3 = 2, then just 3 and 9 are possible, otherwise the leftmost vertical would be divisible by 3.
The probability is ... (read more)