Better Know A Theorem: Languages
I’m grouping together these two short theorems because they deal with languages.
Q: What is special about the following list of primes (in base 10)?
2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049.
A: Given any prime number, you can delete some of its digits to get one of the above primes. You can prove this with A Theorem That Should Be Better Known about well-quasi-ordering; see also Minimal primes by J. Shallit.
English and French Languages
Q: Do the differences between English and French really warrant the division of a country?
A: The languages are both trivial! (and hence equal.) The paper Quotients Homophones des Groupes Libres / Homophonic Quotients of Free Groups is a nicely written short proof of this humourous fact.
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