[FOM] Some questions regarding irrational numbers
dmehkeri at yahoo.ca
Mon Mar 2 12:38:41 EST 2009
> On the other hand, there are plenty of "natural"
> functions of a real variable that we could add into the mix,
> other than exponentiation - should we expect that any of
> these yield some identities that are true but that we cannot
> easily verify?
What about the prime counting function, is that too artificial?
pi(n) is the number of prime numbers less than or equal to n.
Then, lim n->inf (ln(n) - n/pi(n)) was "Legendre's constant". It is now proven to be exactly equal to 1, so it's not your answer. But there may be others along this line - perhaps even equivalent to the Riemann hypothesis!
The assertion that a computable real number is an integer is always going to be a Pi_1 sentence (and conversely given any Pi_1 sentence, we can always artificially construct a computable real number which is an integer if and only if that sentence is true). This is going to rule out some possibilities. But the Riemann hypothesis is equivalent to a Pi_1 sentence, so it's at least possible.
Also let me point out that unlike conjecturing that a given number is rational, or algebraic, we should always be able name the only integer, if any, that a computable number could be equal to - just compute the number to within +/- 0.5
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