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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
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Implications of the disproof of the "climb-to-a-prime" conjecture
Hans said that Conway's point in asking it was that there exist problems easy to state but impossible to prove. The point I took away was that there exist problems that look so hard, nobody has tried …