the golden ratio phi is called the most irrational number because it avoids being close to m/n , or, equivalently, phi*n avoids being close to integers. there is some way to make this precise.
find a number x such that x^n avoids being close to integers, or equivalently, x avoids being close to m^(1/n) for integer m and n. this requires more precise definition.
vaguely similar: Mills's constant, generating prime numbers.
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