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In mathematics, a Lehmer sequence is a generalization of a Lucas sequence.[1]
Algebraic relations[edit]
If a and b are complex numbers with
![{\displaystyle a+b={\sqrt {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13ce25416f07bd0d5593163df101b31357ab5f7c)
![{\displaystyle ab=Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e53151686169f596b930f8c5bc21d7a611ca6c9)
under the following conditions:
Then, the corresponding Lehmer numbers are:
![{\displaystyle U_{n}({\sqrt {R}},Q)={\frac {a^{n}-b^{n}}{a-b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7bb15ca31329fc1ffd792303c569852724f70b5)
for n odd, and
![{\displaystyle U_{n}({\sqrt {R}},Q)={\frac {a^{n}-b^{n}}{a^{2}-b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae7a894f3ec001671f2d40c130b46680771f236)
for n even.
Their companion numbers are:
![{\displaystyle V_{n}({\sqrt {R}},Q)={\frac {a^{n}+b^{n}}{a+b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63a64fad5157cf84ad69ce7ef4ebf5b5b65444c4)
for n odd and
![{\displaystyle V_{n}({\sqrt {R}},Q)=a^{n}+b^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c09c797de23a710a9da071a907f57e45b8182e89)
for n even.
Recurrence[edit]
Lehmer numbers form a linear recurrence relation with
![{\displaystyle U_{n}=(R-2Q)U_{n-2}-Q^{2}U_{n-4}=(a^{2}+b^{2})U_{n-2}-a^{2}b^{2}U_{n-4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52418adc0c0fbcdce4d7f9e1186ed50a50d2e97f)
with initial values
. Similarly the companion sequence satisfies
![{\displaystyle V_{n}=(R-2Q)V_{n-2}-Q^{2}V_{n-4}=(a^{2}+b^{2})V_{n-2}-a^{2}b^{2}V_{n-4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c5da39a306123b42dae553ff5db47ec194a89f)
with initial values
References[edit]
- ^ Weisstein, Eric W. "Lehmer Number". mathworld.wolfram.com. Retrieved 2020-08-11.