On strong Lehmer pseudoprimes in the case of negative discriminant in arithmetic progressions

A. Rotkiewicz
1994 Acta Arithmetica  
1. The Lehmer numbers can be defined as follows: its discriminant is D = L − 4M , and L > 0 and M are rational integers. We can assume without any essential loss of generality that (L, M ) = 1 and M = 0. The Lehmer sequence P k is defined recursively as follows: P 0 = 0, P 1 = 1, and for n ≥ 2, Let V n = (α n + β n )/(α + β) for n odd, and V n = α n + β n for n even denote the nth term of the associated recurring sequence. The associated Lehmer sequence V k can be defined recursively as
more » ... ursively as follows: V 0 = 2, V 1 = 1, and for n ≥ 2, An odd composite number n is a strong Lehmer pseudoprime with parameters L, M (or an sLp for the bases α and β) if (n, DL) = 1, and with δ(n) = n − (DL/n) = d · 2 s , d odd, where (DL/n) is the Jacobi symbol, we have either (i) P d ≡ 0 (mod n), or (ii) V d·2 r ≡ 0 (mod n), for some r with 0 ≤ r < s.
doi:10.4064/aa-68-2-145-151 fatcat:slrfltphyjhjplymyxkq7dfpne