“Lucas Sequence”的概念、定义、习题解题方法及翻译-数学辞典

单词

Lucas Sequence

释义

Lucas Sequence

Let , be Positive Integers. The Roots of

(1)

are

			(2)
			(3)

where

(4)

			(5)
			(6)
			(7)

Then define

			(8)
			(9)

The first few values are therefore

			(10)
			(11)
			(12)
			(13)

The sequences

			(14)
			(15)

are called Lucas sequences, where the definition is usually extended to include

(16)

For , the are the Fibonacci Numbers and are the LucasNumbers. For , the Pell Numbers and Pell-Lucas numbers are obtained. produces the Jacobsthal Numbers and Pell-Jacobsthal Numbers.

The Lucas sequences satisfy the general Recurrence Relations



			(17)


			(18)

Taking

then gives

			(19)
			(20)

Other identities include

			(21)
			(22)
			(23)
			(24)

These formulas allow calculations for large

to be decomposed into a chain in which only four quantities must be kept track ofat a time, and the number of steps needed is

. The chain is particularly simple if

has many 2s in itsfactorization.

The s in a Lucas sequence satisfy the Congruence

(25)

(26)

where

(27)

This fact is used in the proof of the general Lucas-Lehmer Test.

See also Fibonacci Number, Jacobsthal Number, Lucas-Lehmer Test, Lucas Number, Lucas PolynomialSequence, Pell Number, Recurrence Sequence, Sylvester Cyclotomic Number

References

Dickson, L. E. ``Recurring Series; Lucas' , .'' Ch. 17 in History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 393-411, 1952.

Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, pp. 35-53, 1991.

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