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in number theory numbers that in the solution of congruences play a role similar to the role of logarithms in the solution of exponential equations. If p is an odd prime and g is a primitive root with respect to the modulus p, then the index of number a is a number k = ind a such that a = gk(mod p). The properties of indices are
ind ab = ind a + ind b (mod p— 1)
ind (a/b) = ind a— ind b (mod p - 1)
where a/b should be construed as the solution of the congruence bx ≡ a(mod p). Indices can be used to convert a congruence of the form axn ≡ b(mod p) to a linear congruence ind a + n ind x≡ ind b(mod p— 1). Because of the practical use of indices, there are special tables for every prime modulus p (that is not too large). In 1839 the German mathematician C. Jacobi compiled a table of indices for all prime numbers to 1,000. The Soviet mathematician. I. M. Vinogradov has done important studies on the distribution of indices.