Gauss congruence is a property that certain sequences of integers satisfy, including the Lucas numbers and the sequence of divisor sums, and sequences with the property are also known as Dold sequences, Fermat sequences, Newton sequences or realizable sequences. The property is named after Carl Friedrich Gauss, although Gauss never stated it explicitly himself; it can be expressed through a congruence condition built from the Mobius function, and equivalently through the generating function or linear recurrence a sequence satisfies. Sequences obeying Gauss congruence arise naturally in topological dynamics, where they count periodic points of a dynamical system, as well as in algebraic number theory and combinatorics.
Facts
StatementA sequence of integers (a_1, a_2, ...) satisfies Gauss congruence if the sum over the divisors d of n of mu(d) times a_(n/d) is congruent to 0 modulo n for every n >= 1, where mu is the Moebius function. 1 Classification
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Source Gauss congruence, Wikipedia
Source Gauss congruence, Wikipedia
Proved By
Source Gauss congruence, Wikipedia
Sources
1. Gauss congruence, Wikipedia
Lead paragraph and Definition section
In mathematics, Gauss congruence is a property held by certain sequences of integers, including the Lucas numbers and the divisor sum sequence.
In Branch: Algebraic Number Theory, Lead paragraph
Sequences satisfying Gauss congruence naturally occur in the study of topological dynamics, algebraic number theory and combinatorics.
In Branch: Combinatorics, Lead paragraph
Sequences satisfying Gauss congruence naturally occur in the study of topological dynamics, algebraic number theory and combinatorics.
Proved By: Carl Friedrich Gauss, Lead section
The property is named after Carl Friedrich Gauss (1777-1855)
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