Giuga number: Difference between revisions

Latest revision as of 00:27, 23 June 2012

Definition

A composite number $n$ is termed a Giuga number if and only if it satisfies the following equivalent conditions:

For every prime number $p$ dividing $n$ , we have that $p$ divides $(n / p) - 1$ , or equivalently, that $p^{2}$ divides $n - p$ .
We have the following congruence:

$\sum_{i = 1}^{n - 1} i^{φ (n)} \equiv - 1 (\mod n)$

where $φ (n)$ denotes the Euler totient function of $n$ .

Note that all prime numbers satisfy the stated condition but we deliberately exclude them by imposing the restriction of being composite.

Facts

Giuga number is square-free
One of the equivalent formulations of the Agoh-Giuga conjecture is that there is no natural number that is both a Giuga number and a Carmichael number.

Occurrence

Initial examples

30, 858, 1722, 66198, [SHOW MORE]

View list on OEIS

@@ Line 1: / Line 1: @@
 ==Definition==
-A [[composite number]] <math>n</math> is termed a '''Giuga number''' if and only if, for every [[prime number]] <math>p</math> dividing <math>n</math>, we have that <math>p</math> divides <math>(n/p) - 1</math>, or equivalently, that <math>p^2</math> divides <math>n - p</math>.
+A [[composite number]] <math>n</math> is termed a '''Giuga number''' if and only if it satisfies the following equivalent conditions:
+# For every [[prime number]] <math>p</math> dividing <math>n</math>, we have that <math>p</math> divides <math>(n/p) - 1</math>, or equivalently, that <math>p^2</math> divides <math>n - p</math>.
+# We have the following congruence:
+<math>\sum_{i=1}^{n-1} i^{\varphi(n)} \equiv -1 \pmod n</math>
+where <math>\varphi(n)</math> denotes the [[Euler totient function]] of <math>n</math>.
 Note that all [[prime number]]s satisfy the stated condition but we deliberately exclude them by imposing the restriction of being composite.
@@ Line 9: / Line 16: @@
 * [[Giuga number is square-free]]
 * One of the equivalent formulations of the [[Agoh-Giuga conjecture]] is that there is no [[natural number]] that is both a [[Giuga number]] and a [[Carmichael number]].
+==Occurrence==
+===Initial examples===
+<section begin="list"/>[[30]], [[858]], [[1722]], [[66198]], <toggledisplay>2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506</toggledisplay>[[Oeis:A007850|View list on OEIS]]<section end="list"/>