Lucas episode

The Lucas sequence is understood to mean two different things:

• On the one hand the sequence of the Lucas numbers

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, ... (sequence A000032 in OEIS )

where each term in the sequence (from the third) on is the sum of the two preceding ones.

• On the other hand, the two general Lucas sequences and , which are defined depending on the parameters and as those sequences which ${\displaystyle U_{n}(P,Q)}$ ${\displaystyle V_{n}(P,Q)}$ ${\displaystyle P}$ ${\displaystyle Q}$

${\displaystyle U_{0}=0,\quad U_{1}=1}$ or. ${\displaystyle V_{0}=2,\quad V_{1}=P}$

meet and the recursion formulas

${\displaystyle U_{n}=PU_{n-1}-QU_{n-2}\,}$ or. ${\displaystyle V_{n}=PV_{n-1}-QV_{n-2}\,}$

for enough. ${\displaystyle n>1}$

The Lucas sequences are named after the French mathematician Édouard Lucas , who was the first to study them.

Examples

• Be and . Then the following sequence is: ${\displaystyle P=1}$ ${\displaystyle Q=-1}$ ${\displaystyle U_{n}(P,Q)=U_{n}(1,-1)}$

${\displaystyle U_{0}=0,U_{1}=1,}$

${\displaystyle U_{2}=PU_{1}-QU_{0}=1\cdot 1-(-1)\cdot 0=1,}$

${\displaystyle U_{3}=PU_{2}-QU_{1}=1\cdot 1-(-1)\cdot 1=2,}$

${\displaystyle U_{4}=PU_{3}-QU_{2}=1\cdot 2-(-1)\cdot 1=3,}$

${\displaystyle U_{5}=PU_{4}-QU_{3}=1\cdot 3-(-1)\cdot 2=5,\ldots }$

In short one gets the Fibonacci sequence :

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, ... (sequence A000045 in OEIS )

• Be and . Then the following sequence is: ${\displaystyle P=1}$ ${\displaystyle Q=-1}$ ${\displaystyle V_{n}(P,Q)=V_{n}(1,-1)}$

${\displaystyle V_{0}=2,V_{1}=P=1,}$

${\displaystyle V_{2}=PV_{1}-QV_{0}=1\cdot 1-(-1)\cdot 2=3,}$

${\displaystyle V_{3}=PV_{2}-QV_{1}=1\cdot 3-(-1)\cdot 1=4,}$

${\displaystyle V_{4}=PV_{3}-QV_{2}=1\cdot 4-(-1)\cdot 3=7,}$

${\displaystyle V_{5}=PV_{4}-QV_{3}=1\cdot 7-(-1)\cdot 4=11,\ldots }$

In short, you get a sequence that is also called a special Lucas sequence (or even more simply just a Lucas sequence ), namely:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, ... (Follow A000032 in OEIS )

The individual numbers in this sequence are called Lucas numbers , which will be discussed in more detail below .

• Summarized in a table you get for certain start values ​​for and the table in the section special cases . ${\displaystyle P}$ ${\displaystyle Q}$

Explicit formulas

preparation

To determine the terms of the general Lucas sequence, the associated quadratic equation must be solved in advance.

The two solutions and the quadratic equation are required for the explicit formulas . It is this ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle x^{2}-Px+Q=0\ }$

${\displaystyle a={\frac {P}{2}}+{\sqrt {{\frac {P^{2}}{4}}-Q}}={\frac {P+{\sqrt {P^{2}-4Q}}}{2}}}$

and

${\displaystyle b={\frac {P}{2}}-{\sqrt {{\frac {P^{2}}{4}}-Q}}={\frac {P-{\sqrt {P^{2}-4Q}}}{2}}}$

If one of the two complex roots is to be chosen. Which of the two numbers and which is mentioned is irrelevant here. ${\displaystyle P^{2}-4Q<0}$ ${\displaystyle a}$ ${\displaystyle b}$

The parameters and and the values and are interdependent; the reverse applies ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle P=a+b,\quad Q=a\cdot b.}$( Theorem of Vieta )

The formulas for and can be generalized in relation to the powers. The following applies: ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle a^{n}={\frac {V_{n}+U_{n}{\sqrt {P^{2}-4Q}}}{2}}\,}$

${\displaystyle b^{n}={\frac {V_{n}-U_{n}{\sqrt {P^{2}-4Q}}}{2}}\,}$

The general Lucas episodes

If the following applies, or equivalent: if the numbers and are different, the term of the general Lucas sequence is calculated using the following formula: ${\displaystyle P^{2}-4Q\neq 0}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle U_{n}(P,Q)\ }$

${\displaystyle U_{n}(P,Q)={\frac {a^{n}-b^{n}}{a-b}}}$

for everyone . In special cases it applies instead ${\displaystyle n\geq 0}$ ${\displaystyle P^{2}-4Q=0}$

${\displaystyle U_{n}(P,Q)=na^{n-1}=n\left({\frac {P}{2}}\right)^{n-1}.}$

The term in the general Lucas sequence is calculated using the following formula: ${\displaystyle V_{n}(P,Q)\ }$

${\displaystyle V_{n}(P,Q)=a^{n}+b^{n}\ }$

for all ${\displaystyle n\geq 0}$

Relationships between the members

A selection of the relationships between the sequence members is:

• ${\displaystyle U_{2n}=U_{n}\cdot V_{n}\ }$
• ${\displaystyle V_{n}=U_{n+1}-QU_{n-1}\ }$
• ${\displaystyle V_{2n}=V_{n}^{2}-2Q^{n}\ }$
• ${\displaystyle \operatorname {ggT} (U_{m},U_{n})=U_{\operatorname {ggT} (m,n)}}$if ${\displaystyle \operatorname {ggT} (P,Q)=1}$
• ${\displaystyle m\mid n\implies U_{m}\mid U_{n}}$; for all ${\displaystyle U_{m}\neq 1}$

Special cases

There are a few special cases that lead to consequences that play an important role in mathematics and therefore even have their own names:

${\displaystyle P}$	${\displaystyle Q}$	${\displaystyle a}$	${\displaystyle b}$	${\displaystyle U(P,Q)}$	${\displaystyle V(P,Q)}$
${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$	${\displaystyle {\frac {1-{\sqrt {5}}}{2}}}$	${\displaystyle 0,1,1,2,3,5,8,13,21,34,\ldots }$ (Follow A000045 in OEIS ) ( Fibonacci sequence )	${\displaystyle 2,1,3,4,7,11,18,29,47,76,\ldots }$ (Sequence A000032 in OEIS ) ((special) Lucas sequence)
${\displaystyle 1}$	${\displaystyle -2}$	${\displaystyle 2}$	${\displaystyle -1}$	${\displaystyle 0,1,1,3,5,11,21,43,85,171,\ldots }$ (Episode A001045 in OEIS ) ( Jacobsthal episode)	${\displaystyle 2,1,5,7,17,31,65,127,257,511,\ldots }$ (Episode A014551 in OEIS ) (Jacobsthal-Lucas episode)
${\displaystyle 2}$	${\displaystyle -1}$	${\displaystyle 1+{\sqrt {2}}}$	${\displaystyle 1-{\sqrt {2}}}$	${\displaystyle 0,1,2,5,12,29,70,169,408,985,\ldots }$ (Episode A000129 in OEIS ) ( Pell episode )	${\displaystyle 2,2,6,14,34,82,198,478,1154,2786,\ldots }$ (Episode A002203 in OEIS ) (Companion Pell episode, Pell-Lucas episode)
${\displaystyle 3}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 1}$	${\displaystyle 0,1,3,7,15,31,63,127,255,511,\ldots }$ (Sequence A000225 in OEIS ) ( Mersenne number sequence)	${\displaystyle 2,3,5,9,17,33,65,129,257,513,\ldots }$ (Follow A000051 in OEIS ) (Numbers of the form (contain the Fermat numbers )) ${\displaystyle 2^{n}+1}$
${\displaystyle A}$	${\displaystyle -1}$	${\displaystyle {\frac {A+{\sqrt {A^{2}+4}}}{2}}}$	${\displaystyle {\frac {A-{\sqrt {A^{2}+4}}}{2}}}$	Fibonacci polynomials	Lucas polynomials
${\displaystyle 2A}$	${\displaystyle 1}$	${\displaystyle A+{\sqrt {A^{2}-1}}}$	${\displaystyle A-{\sqrt {A^{2}-1}}}$	Chebyshev polynomials of the second kind	Chebyshev polynomials of the first kind, multiplied by ${\displaystyle 2}$
${\displaystyle A+1}$	${\displaystyle A}$	${\displaystyle A}$	${\displaystyle 1}$	${\displaystyle a_{i}=1+a_{i-1}\cdot A}$with repunits to base A ${\displaystyle a_{0}=0}$	${\displaystyle (A^{n}+1)}$ -Episode

But there are also many other special cases that lead to consequences that have an OEIS entry and thus also play a certain role in mathematics. Here are a few examples:

${\displaystyle P}$	${\displaystyle Q}$	${\displaystyle a}$	${\displaystyle b}$	${\displaystyle U(P,Q)}$	${\displaystyle V(P,Q)}$
${\displaystyle 1}$	${\displaystyle 1}$			${\displaystyle 0,1,1,0,-1,-1,0,1,1,0,\ldots }$ (Follow A128834 in OEIS )	${\displaystyle 2,1,-1,-2,-1,1,2,1,-1,-2,\ldots }$ (Follow A087204 in OEIS )
${\displaystyle 1}$	${\displaystyle 2}$			${\displaystyle 0,1,1,-1,-3,-1,5,7,-3,-17,\ldots }$ (Follow A107920 in OEIS )	${\displaystyle 2,1,-3,-5,1,11,9,-13,-31,-5,\ldots }$ (Follow A002249 in OEIS )
${\displaystyle 2}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 0,1,2,3,4,5,6,7,8,9,\ldots }$ (Follow A001477 in OEIS )	${\displaystyle 2,2,2,2,2,2,2,2,2,2,\ldots }$ (Follow A007395 in OEIS )
${\displaystyle 2}$	${\displaystyle 2}$			${\displaystyle 0,1,2,2,0,-4,-8,-8,0,16,\ldots }$ (Follow A009545 in OEIS )	${\displaystyle 2,2,0,-4,-8,-8,0,16,32,32,\ldots }$ (Follow A009545 in OEIS )
${\displaystyle 2}$	${\displaystyle 3}$			${\displaystyle 0,1,2,1,-4,-11,-10,13,56,73,\ldots }$ (Follow A088137 in OEIS )	${\displaystyle 2,2,-2,-10,-14,2,46,86,34,-190,\ldots }$
${\displaystyle 2}$	${\displaystyle 4}$			${\displaystyle 0,1,2,0,-8,-16,0,64,128,0,\ldots }$ (Follow A088138 in OEIS )	${\displaystyle 2,2,-4,-16,-16,32,128,128,-256,-1024,\ldots }$
${\displaystyle 2}$	${\displaystyle 5}$			${\displaystyle 0,1,2,-1,-12,-19,22,139,168,-359,\ldots }$ (Follow A045873 in OEIS )	${\displaystyle 2,2,-6,-22,-14,82,234,58,-1054,-2398,\ldots }$
${\displaystyle 3}$	${\displaystyle -10}$	${\displaystyle 5}$	${\displaystyle -2}$	${\displaystyle 0,1,3,19,87,451,2223,11179,55767,279091,\ldots }$ (Follow A015528 in OEIS )	${\displaystyle 2,3,29,117,641,3093,15689,77997,390881,1952613,\ldots }$
${\displaystyle 3}$	${\displaystyle -5}$			${\displaystyle 0,1,3,14,57,241,1008,4229,17727,74326,\ldots }$ (Follow A015523 in OEIS )	${\displaystyle 2,3,19,72,311,1293,5434,22767,95471,400248,\ldots }$ (Follow A072263 in OEIS )
${\displaystyle 3}$	${\displaystyle -4}$	${\displaystyle 4}$	${\displaystyle -1}$	${\displaystyle 0,1,3,13,51,205,819,3277,13107,52429,\ldots }$ (Follow A015521 in OEIS )	${\displaystyle 2,3,17,63,257,1023,4097,16383,65537,262143,\ldots }$ (Follow A201455 in OEIS )
${\displaystyle 3}$	${\displaystyle -3}$			${\displaystyle 0,1,3,12,45,171,648,2457,9315,35316,\ldots }$ (Follow A030195 in OEIS )	${\displaystyle 2,3,15,54,207,783,2970,11259,42687,161838,\ldots }$ (Follow A172012 in OEIS )
${\displaystyle 3}$	${\displaystyle -2}$			${\displaystyle 0,1,3,11,39,139,495,1763,6279,22363,\ldots }$ (Follow A007482 in OEIS )	${\displaystyle 2,3,13,45,161,573,2041,7269,25889,92205,\ldots }$ (Follow A206776 in OEIS )
${\displaystyle 3}$	${\displaystyle -1}$			${\displaystyle 0,1,3,10,33,109,360,1189,3927,12970,\ldots }$ (Follow A006190 in OEIS )	${\displaystyle 2,3,11,36,119,393,1298,4287,14159,46764,\ldots }$ (Follow A006497 in OEIS )
${\displaystyle 3}$	${\displaystyle 1}$			${\displaystyle 0,1,3,8,21,55,144,377,987,2584,\ldots }$ (Follow A001906 in OEIS )	${\displaystyle 2,3,7,18,47,123,322,843,2207,5778,\ldots }$ (Follow A005248 in OEIS )
${\displaystyle 3}$	${\displaystyle 5}$			${\displaystyle 0,1,3,4,-3,-29,-72,-71,147,796,\ldots }$ (Follow A0190959 in OEIS )	${\displaystyle 2,3,-1,-18,-49,-57,74,507,1151,918,\ldots }$
${\displaystyle 4}$	${\displaystyle -5}$	${\displaystyle 5}$	${\displaystyle -1}$	${\displaystyle 0,1,4,21,104,521,2604,13021,65104,325521,\ldots }$ (Follow A015531 in OEIS )	${\displaystyle 2,4,26,124,626,3124,15626,78124,390626,1953124,\ldots }$ (Follow A087404 in OEIS )
${\displaystyle 4}$	${\displaystyle -3}$			${\displaystyle 0,1,4,19,88,409,1900,8827,41008,190513,\ldots }$ (Follow A015530 in OEIS )	${\displaystyle 2,4,22,100,466,2164,10054,46708,216994,1008100,\ldots }$ (Follow A080042 in OEIS )
${\displaystyle 4}$	${\displaystyle -2}$			${\displaystyle 0,1,4,18,80,356,1584,7048,31360,139536,\ldots }$ (Follow A090017 in OEIS )	${\displaystyle 2,4,20,88,392,1744,7760,34528,153632,683584,\ldots }$
${\displaystyle 4}$	${\displaystyle -1}$			${\displaystyle 0,1,4,17,72,305,1292,5473,23184,98209,\ldots }$ (Follow A001076 in OEIS )	${\displaystyle 2,4,18,76,322,1364,5778,24476,103682,439204,\ldots }$ (Follow A014448 in OEIS )
${\displaystyle 4}$	${\displaystyle 1}$			${\displaystyle 0,1,4,15,56,209,780,2911,10864,40545,\ldots }$ (Follow A001353 in OEIS )	${\displaystyle 2,4,14,52,194,724,2702,10084,37634,140452,\ldots }$ (Follow A003500 in OEIS )
${\displaystyle 4}$	${\displaystyle 2}$			${\displaystyle 0,1,4,14,48,164,560,1912,6528,22288,\ldots }$ (Follow A007070 in OEIS )	${\displaystyle 2,4,12,40,136,464,1584,5408,18464,63040,\ldots }$ (Follow A056236 in OEIS )
${\displaystyle 4}$	${\displaystyle 3}$	${\displaystyle 3}$	${\displaystyle 1}$	${\displaystyle 0,1,4,13,40,121,364,1093,3280,9841,\ldots }$ (Follow A003462 in OEIS )	${\displaystyle 2,4,10,28,82,244,730,2188,6562,19684,\ldots }$ (Follow A034472 in OEIS )
${\displaystyle 4}$	${\displaystyle 4}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 0,1,4,12,32,80,192,448,1024,2304,\ldots }$ (Follow A001787 in OEIS )	${\displaystyle 2,4,8,16,32,64,128,256,512,1024,\ldots }$ (Follow A000079 in OEIS )
${\displaystyle 5}$	${\displaystyle -6}$	${\displaystyle 6}$	${\displaystyle -1}$	${\displaystyle 0,1,5,31,185,1111,6665,39991,239945,1439671,\ldots }$ (Follow A015540 in OEIS )	${\displaystyle 2,5,37,215,1297,7775,46657,279935,1679617,10077695,\ldots }$ (Follow A0274074 in OEIS )
${\displaystyle 5}$	${\displaystyle -3}$			${\displaystyle 0,1,5,28,155,859,4760,26377,146165,809956,\ldots }$ (Follow A015536 in OEIS )	${\displaystyle 2,5,31,170,943,5225,28954,160445,889087,4926770,\ldots }$
${\displaystyle 5}$	${\displaystyle -2}$			${\displaystyle 0,1,5,27,145,779,4185,22483,120785,648891,\ldots }$ (Follow A015535 in OEIS )	${\displaystyle 2,5,29,155,833,4475,24041,129155,693857,3727595,\ldots }$
${\displaystyle 5}$	${\displaystyle -1}$			${\displaystyle 0,1,5,26,135,701,3640,18901,98145,509626,\ldots }$ (Follow A052918 in OEIS )	${\displaystyle 2,5,27,140,727,3775,19602,101785,528527,2744420,\ldots }$ (Follow A087130 in OEIS )
${\displaystyle 5}$	${\displaystyle 1}$			${\displaystyle 0,1,5,24,115,551,2640,12649,60605,290376,\ldots }$ (Follow A004254 in OEIS )	${\displaystyle 2,5,23,110,527,2525,12098,57965,277727,1330670,\ldots }$ (Follow A003501 in OEIS )
${\displaystyle 5}$	${\displaystyle 4}$	${\displaystyle 4}$	${\displaystyle 1}$	${\displaystyle 0,1,5,21,85,341,1365,5461,21845,87381,\ldots }$ (Follow A002450 in OEIS )	${\displaystyle 2,5,17,65,257,1025,4097,16385,65537,262145,\ldots }$ (Follow A052539 in OEIS )
${\displaystyle 8}$	${\displaystyle -9}$	${\displaystyle 9}$	${\displaystyle -1}$	${\displaystyle 0,1,8,73,656,5905,53144,478297,4304672,38742049,\ldots }$ (Follow A015577 in OEIS )	${\displaystyle 2,8,82,728,6562,59048,531442,4782968,43046722,387420488,\ldots }$

The general Lucas sequences U (P, Q) , V (P, Q) and the prime numbers

The general Lucas sequences and have a special property for integer parameters and with regard to the divisibility by prime numbers. This property was used for methods to determine the primality of a number (see also Lucas-Lehmer test ). ${\displaystyle U(P,Q)\,}$ ${\displaystyle V(P,Q)\,}$ ${\displaystyle P}$ ${\displaystyle Q}$

The sequences U (P, Q)

The following applies to all Lucas episodes : ${\displaystyle U_{n}(P,Q)={\frac {a^{n}-b^{n}}{a-b}}}$

If p is a prime number, then it is divisible by p . ${\displaystyle U_{p}(P,Q)-\left({\frac {D}{p}}\right)}$

It is the Legendre symbol . ${\displaystyle \left({\frac {D}{p}}\right)}$

There are also composite numbers that meet this condition. These numbers are called Lucas pseudoprimes .

The sequences V (P, Q)

The following applies to all Lucas episodes : ${\displaystyle V_{n}(P,Q)=a^{n}+b^{n}\ }$

If p is a prime number, then by divisible. ${\displaystyle V_{p}(P,Q)-P\ }$ ${\displaystyle p}$

A composite number that fulfills this condition (in the case of and ) is called a Fibonacci pseudoprime number . ${\displaystyle P>0}$ ${\displaystyle Q=\pm 1}$

The divisibility condition is particularly interesting for the sequence . The following applies to this sequence: ${\displaystyle V_{n}(3,2)=a^{n}+b^{n}=2^{n}+1\ }$

If is a prime number then: divides . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle 2^{n}+1-3=2^{n}-2\ }$

This is a special form of the little Fermatschen theorem .

The same applies here . ${\displaystyle a^{p}\equiv a\mod p}$ ${\displaystyle V_{p}(a+1,a)\equiv V_{1}(a+1,a)\mod p}$

The special Lucas series

The generally known Lucas-Sequence Sequence of Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, ... can be other than by the recursion with the initial values and also generate, as follows: ${\displaystyle L_{n}}$ ${\displaystyle L_{n}=L_{n-1}+L_{n-2}}$ ${\displaystyle L_{0}=2}$ ${\displaystyle L_{1}=1}$

1. As mentioned in the general case for the consequences , about Binet's formula (after Jacques Philippe Marie Binet ): ${\displaystyle V_{n}}$

   ${\displaystyle L_{n}=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}+\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}}$, there and applies. By the way, a is the golden number . ${\displaystyle a={\frac {1+{\sqrt {5}}}{2}}}$ ${\displaystyle b={\frac {1-{\sqrt {5}}}{2}}}$ ${\displaystyle \Phi }$

2. Another recursive formula (with Gaussian brackets ):

   ${\displaystyle L_{n+1}=\left\lfloor {\frac {L_{n}(1+{\sqrt {5}})+1}{2}}\right\rfloor }$

3. As the sum of two terms of the Fibonacci sequence :

   ${\displaystyle L_{n}=f_{n-1}+f_{n+1}\ }$ .

Alternatively, you can also write according to 1) . As for the amount of is always less than 0.5, there is the property that the th ( ) Luca number the rounded value of the Golden number to the power of equal to: . ${\displaystyle L_{n}=\Phi ^{n}+(1-\Phi )^{n}}$ ${\displaystyle n>1}$ ${\displaystyle (1-\Phi )^{n}}$ ${\displaystyle n}$ ${\displaystyle n>1}$ ${\displaystyle n}$ ${\displaystyle L_{n}=\left\lfloor {\Phi ^{n}+{\frac {1}{2}}}\right\rfloor }$

Reciprocal series

The limit of the absolutely converging reciprocal series of special Lucas numbers

• ${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{L_{2^{n}}}}}$

is irrational .

Lucas prime numbers

A Lucas prime is a Lucas number that is prime. The smallest Lucas prime numbers are:

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688,846,502,588,399, 32361122672259149, 412670427844921037470771 ... (sequence A005479 in OEIS )

For this Lucas primes the index of the following: ${\displaystyle n}$ ${\displaystyle L_{n}}$

0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849, ... (sequence A001606 in OEIS )

Example:

It is and . Thus is a prime number. In fact, the index appears in the list above in the 5th position because it leads to the fifth smallest Lucas prime number . ${\displaystyle L_{6}=18}$ ${\displaystyle L_{5}=11}$ ${\displaystyle L_{7}=L_{6}+L_{5}=18+11=29\in \mathbb {P} }$ ${\displaystyle n=7}$ ${\displaystyle L_{7}=29}$

The following two properties apply to Lucas prime numbers:

• If is a prime, then the index is either equal to, or even prime, or a power of two . ${\displaystyle L_{n}}$ ${\displaystyle n}$ ${\displaystyle 0}$
• ${\displaystyle L_{2^{m}}}$is a prime number for . No other prime numbers are obtained for any other known values ​​of . ${\displaystyle m\in \{1,2,3,4\}}$ ${\displaystyle m}$

It is believed that there are infinitely many Lucas primes.

See also

• linear difference equation

literature

• Paulo Ribenboim : The world of prime numbers . Springer Verlag, 1996

Web links

• Eric W. Weisstein : Lucas Number . In: MathWorld (English).
• Eric W. Weisstein : Lucas Sequence . In: MathWorld (English).

Individual evidence

1. ↑ See Ribenboim: The World of Prime Numbers , pp. 44–70.
2. ↑ See the chapter already mentioned in the book by Ribenboim.
3. ↑ Paulo Ribenboim: My Numbers, My Friends: Highlights of Number Theory. Springer textbook, 2009, ISBN 978-3-540-87955-8 , p. 323.
4. ↑ ^{a b} Chris K. Caldwell: Lucas prime. Prime Pages, accessed March 1, 2020 .