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Pell-Lucas Number


The Pell-Lucas numbers are the V_ns in the Lucas sequence with P=2 and Q=-1, and correspond to the Pell-Lucas polynomial Q_n(1).

The Pell-Lucas number Q_n is equal to

 Q_n=F_(n-1)(2)+F_(n+1)(2),
(1)

where F_n(x) is a Fibonacci polynomial.

The Pell-Lucas and Pell numbers satisfy the recurrence relation

 Q_n=2Q_(n-1)+Q_(n-2)
(2)

with initial conditions Q_0=Q_1=2 for the Pell-Lucas numbers and P_0=0 and P_1=1 for the Pell numbers.

The nth Pell-Lucas number is explicitly given by the Binet-type formulas

 Q_n=(1-sqrt(2))^n+(1+sqrt(2))^n.
(3)

The nth Pell-Lucas number is given by the binomial sums

 Q_n=2sum_(k=0)^(|_n/2_|)(n; 2k)2^k,
(4)

The Pell-Lucas numbers satisfy the identities

Q_n^2=4[2P_n^2+(-1)^n]
(5)
Q_(2n)=Q_n^2-2(-1)^n.
(6)

For n=0, 1, ..., the Pell-Lucas numbers are 2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, ... (OEIS A002203). As can be seen, they are always even.

For a Pell-Lucas number Q_n/2 to be prime, it is necessary that n be either prime or a power of 2. The indices of Q_n/2 that are (probable) primes are 2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, 6689, 8087, 9679, 28753, 79043, 129127, 145969, 165799, 168677, 170413, 172243, 278321, ... (OEIS A099088). The largest proven prime has index 9679 and 3705 decimal digits (http://primes.utm.edu/primes/page.php?id=27783). These indices k are a superset via 2k^'+1 of the indices k^' of prime NSW numbers. The following table summarizes the largest known Pell-Lucas (probable) primes.

ndecimal digitsdiscovererdate
12912749427E. W. WeissteinMay 19, 2006
14596955874E. W. WeissteinAug. 29, 2006
16579963464E. W. WeissteinNov. 16, 2006
16867764566E. W. WeissteinNov. 26, 2006
17041365230E. W. WeissteinDec. 10, 2006
17224365931E. W. WeissteinJan. 15, 2007
278321106535R. PriceDec. 7, 2018

See also

Brahmagupta Polynomial, Integer Sequence Primes, NSW Number, Pell Number

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References

Ram, R. "Pell Numbers Formulae." http://users.tellurian.net/hsejar/maths/pell/.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 53-57, 1996.Sloane, N. J. A. Sequences A002203/M0360 and A099088 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Pell-Lucas Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pell-LucasNumber.html

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