Editing Carmichael number (section)

== Higher-order Carmichael numbers ==
Carmichael numbers can be generalized using concepts of [[abstract algebra]].

The above definition states that a composite integer ''n'' is Carmichael
precisely when the ''n''th-power-raising function ''p''<sub>''n''</sub> from the [[ring (mathematics)|ring]] '''Z'''<sub>''n''</sub> of integers modulo ''n'' to itself is the identity function. The identity is the only '''Z'''<sub>''n''</sub>-[[algebra over a field|algebra]] [[endomorphism]] on '''Z'''<sub>''n''</sub> so we can restate the definition as asking that ''p''<sub>''n''</sub> be an algebra endomorphism of '''Z'''<sub>''n''</sub>.
As above, ''p''<sub>''n''</sub> satisfies the same property whenever ''n'' is prime.

The ''n''th-power-raising function ''p''<sub>''n''</sub> is also defined on any '''Z'''<sub>''n''</sub>-algebra '''A'''. A theorem states that ''n'' is prime if and only if all such functions ''p''<sub>''n''</sub> are algebra endomorphisms.

In-between these two conditions lies the definition of '''Carmichael number of order m''' for any positive integer ''m'' as any composite number ''n'' such that ''p''<sub>''n''</sub> is an endomorphism on every '''Z'''<sub>''n''</sub>-algebra that can be generated as '''Z'''<sub>''n''</sub>-[[module (mathematics)|module]] by ''m'' elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.

=== An order-2 Carmichael number ===
According to Howe, 17 · 31 · 41 · 43 · 89 · 97 · 167 · 331 is an order 2 Carmichael number. This product is equal to 443,372,888,629,441.<ref>{{cite journal |author = Everett W. Howe |title=Higher-order Carmichael numbers |journal=Mathematics of Computation |date=October 2000 |volume=69 |issue=232 |pages=1711–1719 |arxiv=math.NT/9812089 |jstor=2585091 |doi=10.1090/s0025-5718-00-01225-4|bibcode=2000MaCom..69.1711H |s2cid=6102830 }}</ref>

=== Properties ===
Korselt's criterion can be generalized to higher-order Carmichael numbers, as shown by Howe.

A heuristic argument, given in the same paper, appears to suggest that there are infinitely many Carmichael numbers of order ''m'', for any ''m''. However, not a single Carmichael number of order 3 or above is known.