Lehmer number

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

Algebraic relations[edit]

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.

Recurrence[edit]

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.