indices


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indices

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A plural of "index".

Indices

 

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.

REFERENCE

Vinogradov, I. M. Osnovy teorii chisel, 8th ed. Moscow, 1972.
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