Lehmer number

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

[edit] Algebraic relations

If a and b are complex numbers with

a + b = \sqrt{R}
ab = Q

under the following conditions:

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.

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