An infinite class of planar configurations is constructed with distinct prime-field characteristic sets (i.e., configurations represented over a finite set of prime fields but over fields of no other characteristic). It is shown that if p p is sufficiently large, then every subset of k k primes between p p and f ( p , k ) f(p,k) forms such a set (where f ( p , k ) = 2 [ ( p − A k 3 / 2 ) / B k 3 / 2 ] f(p,k) = {2^{[(\sqrt p - A{k^{3/2}})/B{k^{3/2}}]}} for constants A A and B B ). In particular, for every positive integer k k , there exist infinitely many planar matroid configurations C i , k {C_{i,k}} with | χ p f ( C i , k ) | = k \left | {{\chi _{pf}}({C_{i,k}})} \right | = k (where χ p f ( C ) {\chi _{pf}}(C) denotes the prime-field characteristic set of C C ). We also give a result concerning cofinite prime-field characteristic sets.