Lehmer Number

In mathematics, a Lehmer sequence $U_n(\sqrt R, Q)$ or $V_n(\sqrt R, Q)$ is a generalization of a Lucas sequence $U_n(P, Q)$ or $V_n(P, Q)$, allowing the square root of an integer ''R'' in place of the integer ''P''.

To ensure that the value is always an integer, every other term of a Lehmer sequence is divided by compared to the corresponding Lucas sequence. That is, when ''R'' = ''P''² the Lehmer and Lucas sequences are related as:
:$\begin P\,U_(\sqrt,Q) &= U_(P,Q) & U_(\sqrt,Q) &= U_(P,Q) \\ V_(\sqrt,Q) &= V_(P,Q) & P\,V_(\sqrt,Q) &= V_(P,Q) \end$

Algebraic relations

If ''a'' and ''b'' are complex numbers with

:$a + b = \sqrt$
:$ab = Q$

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:

:$U_n(\sqrt,Q) = \frac$

for ''n'' odd, and

:$U_n(\sqrt,Q) = \frac$
for ''n'' even.

Their companion numbers are:

:$V_n(\sqrt,Q) = \frac$

for ''n'' odd and

:$V_n(\sqrt,Q) = a^n+b^n$

for ''n'' even.

Recurrence

Lehmer numbers form a linear recurrence relation with

:$U_n = (R-2Q)U_-Q^2U_ = (a^2+b^2)U_-a^2b^2U_$

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

:$V_n = (R-2Q)V_-Q^2V_ = (a^2+b^2)V_-a^2b^2V_$

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

All Lucas sequence recurrences apply to Lehmer sequences if they are divided into cases for even and odd ''n'' and appropriate factors of are incorporated. For example,

:$\begin U_(\sqrt R,Q) &= \phantom U_(\sqrt R,Q) - Q\, U_(\sqrt R,Q) & U_(\sqrt R,Q) &= R\, U_(\sqrt R,Q) - Q\, U_(\sqrt R,Q) \\ V_(\sqrt R,Q) &= R\, V_(\sqrt R,Q) - Q\, V_(\sqrt R,Q) & V_(\sqrt R,Q) &= \phantom V_(\sqrt R,Q) - Q\, V_(\sqrt R,Q) \end$

References

Integer sequences

{{numtheory-stub