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)
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where is a Fibonacci polynomial.
The Pell-Lucas and Pell numbers satisfy the recurrence relation
(2)
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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)
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The th Pell-Lucas number is given by the binomial sums
(4)
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The Pell-Lucas numbers satisfy the identities
(5)
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(6)
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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 |