Reststelsel

In de elementaire getaltheorie is een reststelsel of restsysteem modulo het positieve gehele getal een verzameling getallen uit verschillende restklassen modulo . Een reststelsel bestaat dus uit een aantal getallen waarvan er geen twee congruent zijn modulo .

Er wordt verschil gemaakt tussen volledige reststelsels en gereduceerde reststelsels.

Volledig reststelsel

Een reststelsel heet een volledig reststelsel modulo als het stelsel van iedere restklasse een element bevat. Een volledig reststelsel heeft dus precies elementen.

Gereduceerd reststelsel

Een reststelsel heet een gereduceerd reststelsel als het bestaat uit getallen die met relatief priem zijn. Het aantal elementen is gelijk aan . Dus:

  1. voor elke ;
  2. ;
  3. Geen twee elementen van zijn congruent modulo .

Hierin is de indicator- of totientfunctie.

Een dergelijk gereduceerd reststelsel modulo ontstaat uit een volledig reststelsel door alle getallen die niet relatief priem zijn met weg te laten.

Voorbeeld

Een volledig reststelsel modulo 12 is bijvoorbeeld {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. Daarin zijn 1, 5, 7 en 11 de enige getallen die relatief priem zijn met 12. Zij vormen een gereduceerd reststelsel modulo 12: {1,5,7,11}. De kardinaliteit van dit stelsel is 4 en gelijk aan de indicator van 12: .

Andere gereduceerde reststelsels modulo 12 zijn { 13, 17, 19, 23 } en { - 11, -7, -5, -1 }.

Eigenschappen

Als een gereduceerd reststelsel is en , dan is

.

Ieder getal in een gereduceerd reststelsel modulo is een voortbrenger van de additieve groep van gehele getallen modulo .

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