Lucas Pseudoprime
المؤلف:
Baillie, R. and Wagstaff, S. S. Jr.
المصدر:
"Lucas Pseudoprimes." Math. Comput. 35
الجزء والصفحة:
...
24-1-2021
1297
Lucas Pseudoprime
When 
and 
are integers such that 
, define the Lucas sequence ![<span style=]()
{U_k}" src="https://mathworld.wolfram.com/images/equations/LucasPseudoprime/Inline4.gif" style="height:15px; width:26px" /> by
for 
, with 
and 
the two roots of 
. Then define a Lucas pseudoprime as an odd composite number 
such that 
, the Jacobi symbol 
, and 
.
The congruence 
holds for every prime number 
, where 
is a Lucas number. However, some composites also satisfy this congruence. The Lucas pseudoprimes corresponding to the special case of the Lucas numbers 
are those composite numbers 
such that 
. The first few of these are 705, 2465, 2737, 3745, 4181, 5777, 6721, ... (OEIS A005845).
The Wolfram Language implements the multiple Rabin-Miller test in bases 2 and 3 combined with a Lucas pseudoprime test as the primality test in the function PrimeQ[n].
REFERENCES:
Baillie, R. and Wagstaff, S. S. Jr. "Lucas Pseudoprimes." Math. Comput. 35, 1391-1417, 1980.
Bruckman, P. S. "Lucas Pseudoprimes are Odd." Fib. Quart. 32, 155-157, 1994.
Ribenboim, P. "Lucas Pseudoprimes (lpsp(
))." §2.X.B in The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, p. 129, 1996.
Sloane, N. J. A. Sequence A005845/M5469 in "The On-Line Encyclopedia of Integer Sequences."
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