Editing Carmichael's theorem (section)

==Statement==
Given two relatively prime integers ''P'' and ''Q'', such that <math>D=P^2-4Q>0</math> and {{math|''PQ''&nbsp;≠&nbsp;0}}, let {{math|''U<sub>n</sub>''(''P'',&nbsp;''Q'')}} be the [[Lucas sequence]] of the first kind defined by
:<math>\begin{align}
U_0(P,Q)&=0, \\
U_1(P,Q)&=1, \\
U_n(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q) \qquad\mbox{  for }n>1.
\end{align}
</math>

Then, for ''n''&nbsp;≠&nbsp;1,&nbsp;2,&nbsp;6, ''U<sub>n</sub>''(''P'',&nbsp;''Q'') has at least one prime divisor that does not divide any ''U<sub>m</sub>''(''P'',&nbsp;''Q'') with ''m''&nbsp;&lt;&nbsp;''n'', except ''U''<sub>12</sub>(±1,&nbsp;−1)&nbsp;=&nbsp;±F(12)&nbsp;=&nbsp;±144.
Such a prime&nbsp;''p'' is called a ''characteristic factor'' or a ''primitive prime divisor'' of ''U<sub>n</sub>''(''P'',&nbsp;''Q'').
Indeed, Carmichael showed a slightly stronger theorem: For ''n''&nbsp;≠&nbsp;1,&nbsp;2, 6, ''U<sub>n</sub>''(''P'',&nbsp;''Q'') has at least one primitive prime divisor not dividing ''D''<ref>In the definition of a primitive prime divisor&nbsp;''p'', it is often required that ''p'' does not divide the discriminant.</ref> except ''U''<sub>3</sub>(±1,&nbsp;−2)&nbsp;=&nbsp;3, ''U''<sub>5</sub>(±1,&nbsp;−1)&nbsp;=&nbsp;F(5)&nbsp;=&nbsp;5, or ''U''<sub>12</sub>(1,&nbsp;−1)&nbsp;=&nbsp;−''U''<sub>12</sub>(−1,&nbsp;−1)&nbsp;=&nbsp;F(12)&nbsp;=&nbsp;144.

In Camicharel's theorem, ''D'' should be greater than 0; thus the cases ''U''<sub>13</sub>(1,&nbsp;2), ''U''<sub>18</sub>(1,&nbsp;2) and ''U''<sub>30</sub>(1,&nbsp;2), etc. are not included, since in this case ''D''&nbsp;=&nbsp;−7&nbsp;<&nbsp;0.