The elementary numbers are the complex numbers which can be implicitly or explicitly defined by starting with the rationale and using addition, subtraction, multiplication and exponentiation. More explicitly, an elementary point is a non-singular solution of n equations, involving exponential polynomials, in n unknowns; and an elementary number is obtained by applying an exponential polynomial to an elementary point. The elementary constant problem is the problem of deciding whether or not a number described in this way is actually zero. Schanuel’s conjecture is used and a concept of local primality developed for exponential ideals to attempt to solve the problem. The result is that part of the problem, for the smallest algebraically closed field which is also closed under ex and logx, is solved in the sense that a program would be written which would never give the wrong answer, and would only fail to terminate in cases in which it also discovered a counterexample to Schanuel’s conjecture.