# [FOM] Difficult?

Lucas Wiman lrwiman at ilstu.edu
Wed Apr 30 13:59:38 EDT 2003

``` > This still does not answer the original question about hardness. A
> hardness result would probably be quite difficult to show as it would
> effectively imply that there is no poly-time algorithm for computing
f(n).

Unless I'm misunderstanding you, then the question of poly-time
algorithm for calculating pi(n) mod 2 is not relevant to Friedman's
question about how many symbols this calculation would take in ZFC.
There is, for example, a poly-time algorithm for determining whether
Mersenne numbers are prime (either Agrawal et al's recent algorithm or
the more classical Lucas-Lehmer test).  Yet we have no reason (to my
knowledge) to suspect that if 2^(2^127-1)-1 is prime, then it would
require fewer than 2^100 symbols to prove it in ZFC.  Even a linear time
algorithm would require O(2^127) symbols to do it.

(Aside:  The primality of this "double Mersenne prime" has been looked
at, and to my knowledge no factor is known of it.  The reason it was
looked at is as follows:
2 is prime
3=2^2-1 is prime
7=2^3-1 is prime
127=2^7-1 is prime
2^127-1 is prime
2^(2^127-1)-1 must be prime too, right?)

- Lucas Wiman

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