Rédei theorem

This article deals with the theorem of elementary number theory; on the theorem in graph theory see Rédei's theorem (graph theory) .

The set of Rédei is a theorem of elementary number theory , a sub-region of mathematics . It goes back to the Hungarian mathematician László Rédei and is closely related to the Euler-Fermat theorem , which it even leads to.

Formulation of the sentence

The Rédeic sentence says the following:

For every natural number and every integer is ${\displaystyle m\geq 2}$ ${\displaystyle a}$

${\displaystyle a^{m}-a^{m-\phi (m)}}$

by divisible , wherein the number of natural numbers below stands which to prime are. ${\displaystyle m}$ ${\displaystyle \phi (m)}$ ${\displaystyle m}$ ${\displaystyle m}$

It is therefore always the congruence

${\displaystyle a^{m}\equiv a^{m-\phi (m)}\mod {m}}$

valid.

literature

• H. Alzer: The Euler-Fermat theorem . In: International Journal of Education in Mathematics, Science and Technology . tape 18 , 1987, pp. 635-636 .
• József Sándor, Borislav Crstici: Handbook of Number Theory. II . Chapter 3: The Many Facets of Euler's Totient. Kluwer Academic Publishers, Dordrecht / Boston / London 2004, ISBN 1-4020-2546-7 ( MR2119686 ).
• Wacław Sierpiński : Elementary Theory of Numbers (=  North-Holland Mathematical Library . Volume 31 ). 2nd revised and expanded edition. North-Holland ( inter alia), Amsterdam ( inter alia ) 1988, ISBN 0-444-86662-0 ( MR0930670 ).

Individual evidence

1. ^ Wacław Sierpiński: Elementary Theory of Numbers . 1988, pp. 261-262
2. ↑ József Sándor, Borislav Crstici: Handbook of Number Theory. II . 2004, pp. 189-190, 208
3. ↑ The arithmetic function that occurs here is Euler's Phi function named after Leonhard Euler .