Lucas sequence
Under the Lucas sequence is understood to mean two different things:
- On the one hand the result of Lucas numbers
- On the other hand, the two general Lucas sequences and which are defined depending on the parameters and, as those sequences which
The Lucas sequences are named after the French mathematician Edouard Lucas, who has studied the first with them.
- 4.1 The consequences of U ( P, Q)
- 4.2 Follow V ( P, Q)
Explicit formulas
Preparation
To determine the general Lucas sequence of sequence elements the associated quadratic equation must be solved in preparation.
For the explicit formulas and the two solutions of the quadratic equation are needed. These are
And
If, as is one of the two complex roots to choose. Which of the two numbers and what is called, is this not a concern.
And the parameters, and and the values are dependent on each other, it is vice versa
The formulas for a and b can be generalized in terms of magnitude. Namely, the following applies:
The general Lucas sequences
If applies, or equivalently: if the numbers are different and so is calculated, the member of the general Lucas sequence according to the following formula:
For everyone. Be instead in the special case
The general Lucas sequence The link is calculated according to the following formula:
For all
Relationships between the sequence elements
A selection of the relations between the sequence elements is:
- ; for all
Special cases
General Lucas sequences U ( P, Q ), V ( P, Q) and the prime numbers
The general Lucas sequences and have for integer parameters and a specific property in terms of divisibility by prime numbers. This property has been applied for method of determining the primality of a number ( see also Lucas- Lehmer test ).
The consequences of U ( P, Q)
For all Lucas sequences applies:
It is the Legendre symbol.
There are also composite numbers that satisfy this condition. These numbers are called Lucas pseudoprimes.
The consequences V ( P, Q)
For all Lucas sequences applies:
A composite number (in the case of P> 0 ) satisfying this condition is the pseudo Fibonacci prime.
Particularly interesting is the Teilbarkeitsbedingung for the result. Namely applies for this episode:
This is a special form of Fermat's little theorem.
Analogous to apply here.
The special Lucas sequence
The generally known Lucas - result result of Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, ... can be generated not only by the recursion with the initial values , and also as follows:
1) As mentioned in the general case for the consequences of the formula of Binet (after Jacques Philippe Marie Binet ):
There and apply. a way, is the golden number.
2 ) Another recursive formula ( with floor function ):
3) The sum of the Fibonacci sequence of two terms: