  In 1990, Mermin presented a n player game that is won with certainty using n
spin-1/2 particles in a GHZ state whilst no classical strategy (or local
theory) can win with probability higher than ${1/2} + \frac{1}{2^{\lceil n/2
\rceil}}$ (which is larger than 1/2). This article first introduces a class of
arithmetic games containing Mermin's and gives a quantum algorithm based on a
generalized n party GHZ state that wins those games with certainty. It is then
proved for a subclass of those games where each player is given a single bit of
input that no classical strategy can win with a probability that is
asymptotically larger than 1.6 times the inverse of the square root of n, thus
giving a new and stronger Bell inequality.
