The Palindrome Block Theorem

For coprime positive integers p, q with p is equivalent to 1 pmod q, this note gives a complete, closed-form description of representability for N = pA + qB, A, B is greater than or equal to 0, below the Frobenius bound (p minus 1)(q minus 1). Writing N = dq + r with 0 is less than or equal to r < q and setting m = (p minus 1)/q, we show that N is representable if and only if d is greater than or equal to mr. This criterion decomposes the sub-Frobenius region into exactly q minus 1 levels, each made up of m consecutive rows in which a block of j representable values is followed by q minus j gaps, for j = 1, and so on , q minus 1. As a corollary, summing the representable cells across all levels reproduces the classical Sylvester-Frobenius genus formula g = (p minus 1)(q minus 1)/2 through a direct, level-by-level combinatorial count, giving an alternative to the usual symmetry argument. The result is illustrated with five worked examples covering different values of m and q, including the boundary case m=1 and the minimal case q=2.