The Pell-Lucas numbers 
, for nonnegative integers 
, are the 
terms in the Lucas sequence
with 
and 
,
and correspond to the Pell-Lucas polynomial 
.
For 
,
the Pell-Lucas number 
is equal to
 |
(1)
|
where 
is a Fibonacci polynomial.
The Pell-Lucas and Pell numbers satisfy the recurrence relation
 |
(2)
|
for 
,
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 formula
 |
(3)
|
The 
th
Pell-Lucas number is given by the binomial sums
 |
(4)
|
where 
is the floor function.
The Pell-Lucas numbers satisfy the identities
 |  | ![4[2P_n^2+(-1)^n]](/images/equations/Pell-LucasNumber/Inline19.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 half a Pell-Lucas number, 
, to be prime, it is necessary
that 
be either prime or a power
of 2. The indices of 
that are primes or 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, 552283, ... (OEIS A099088). The term
has 3705 decimal digits and is listed as a probable prime (https://t5k.org/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 probable primes
of the form 
.
 | 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 |
 |  | R. Price | reported Dec. 2023 |
