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
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Gauss congruence, Wikipedia
Lead paragraph and Definition sectionQuote, 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.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.