Lehmer number

In mathematics, a Lehmer number is a generalization of a Lucas sequence.
Algebraic relations

If a and b are complex numbers with

\( a + b = \sqrt{R} \)

ab = Q

under the following conditions:

Q and R are relatively prime nonzero integers
a/b is not a root of unity.

Then, the corresponding Lehmer numbers are:

\( U_n(\sqrt{R},Q) = \frac{a^n-b^n}{a-b} \)

for n odd, and

\( U_n(\sqrt{R},Q) = \frac{a^n-b^n}{a^2-b^2} \)

for n even.

Their companion numbers are:

\( V_n(\sqrt{R},Q) = \frac{a^n+b^n}{a+b} \)

for n odd and

\( V_n(\sqrt{R},Q) = a^n+b^n \)

for n even.
Recurrence

Lehmer numbers form a linear recurrence relation with

\( U_n=(R-2Q)U_{n-2}-Q^2U_{n-4}=(a^2+b^2)U_{n-2}-a^2b^2U_{n-4} \)

with initial values \( U_0=0,U_1=1,U_2=1,U_3=R-Q=a^2+ab+b^2 \). Similarly the companions sequence satisfies

\( V_n=(R-2Q)V_{n-2}-Q^2V_{n-4}=(a^2+b^2)V_{n-2}-a^2b^2V_{n-4} \)

with initial values \( V_0=2,V_1=1,V_2=R-2Q=a^2+b^2,V_3=R-3Q=a^2-ab+b^2. \)

1 Rising and falling factorials
2 Identities and relations
3 Table of values
4 See also

Mathematics Encyclopedia