Strong Lucas Pseudoprime

Let and be Lucas sequences generated by and , and define

(1)

Let be an odd composite number with , and with odd and , where is the Legendre symbol. If

(2)

or

(3)

for some with , then is called a strong Lucas pseudoprime with parameters .

A strong Lucas pseudoprime is a Lucas pseudoprime to the same base. Arnault (1997) showed that any composite number is a strong Lucas pseudoprime for at most 4/15 of possible bases (unless is the product of twin primes having certain properties).