If F(n) is the nth Fermat prime, then

sum(n odd) 1/F(n)

and

sum(n even) 1/F(n)

and

product(n odd) (1-1/F(n))

and

product(n even) (1-1/F(n))

each are irrational... IFF there are an infinite number of Fermat primes

of the kind in that expression.

One must admit, though, that this is a pretty silly result, for

several reasons. I guess it is best posed as a puzzle, then :)