The Pell-Lucas numbers are the 
s in the Lucas sequence
with 
and 
,
and correspond to the Pell-Lucas polynomial 
.
The Pell-Lucas number 
is equal to
 |
(1)
|
where 
is a Fibonacci polynomial.
The Pell-Lucas and Pell numbers satisfy the recurrence relation
 |
(2)
|
with initial conditions 
for the Pell-Lucas numbers and 
and 
for the Pell numbers.
The 
th
Pell-Lucas number is explicitly given by the Binet-type formulas
 |
(3)
|
The 
th
Pell-Lucas number is given by the binomial sums
 |
(4)
|
The Pell-Lucas numbers satisfy the identities
 |  | ![4[2P_n^2+(-1)^n]](/images/equations/Pell-LucasNumber/Inline14.svg) |
(5)
|
 |  |  |
(6)
|
For 
,
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 
to be prime, it is necessary that 
be either prime or a power of 2. The indices of 
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 
are a superset via 
of the indices 
of prime NSW numbers. The following table summarizes
the largest known Pell-Lucas (probable) primes.
 | decimal digits | discoverer | date |
 |  | E. W. Weisstein | May 19, 2006 |
 |  | E. W. Weisstein | Aug. 29, 2006 |
 |  | E. W. Weisstein | Nov. 16, 2006 |
 |  | E. W. Weisstein | Nov. 26, 2006 |
 |  | E. W. Weisstein | Dec. 10, 2006 |
 |  | E. W. Weisstein | Jan. 15, 2007 |
 |  | R. Price | Dec. 7, 2018 |
