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Let U,, = (a' /')/(a /-3) for n odd and Un = (a" /")/(a2 2) for even n, where a and /3 are distinct roots of the trinomial f(z) = L Z + Q and L > 0 and Q are rational integers. Un is the n th Lehmer number connected with f(z ). Let [ = (a + /3X)/(a ? /3) for n odd, and [ an + /31 for n even denote the nith term of the associated recurring sequence. An odd composite number n is a strong Lehmner pseudoprime with parameters L, Q (or slepsp(L, Q)) if (n, DQ) = 1, where D = L 4Q * 0, and with 8(n) = n (DL/n) d 2S, d odd, where (DL/n) is the Jacobi symbol, we have either Ud 0 (mod n) or kd 2' 0 (mod n), for some r with 0 0. Then every arithmetic progression ax + b, where a, b are relatively prime integers, contains an infinite number of odd (composite) strong Lehmer pseudoprimes with parameters L, Q. Some new tests for primality are also given. 1. First we recall the definitions of Euler pseudoprimes, which have been introduced (see Pomerance, Selfridge, Wagstaff [5]) because they are rarer than ordinary pseudoprimes. An odd composite number n is an Euler pseudoprime to base c (or epsp(c)) if (c, n) = 1 and (1) ~ ~ ~ ~ ~~~~n-1/2 (l) C(n-1)/2 ~c-(mod n), where (cln) is the Jacobi symbol (see also Lehmer [4]). An odd composite n is a strong pseudoprime for the base c (or spsp(c)) if, with n 1 = d 2, d odd, we have (2) cd-1 (mod n) or Cd2_ -1 (mod n) for some r with 0 < r < s. Any prime p with (p, c) = 1 satisfies one or the other term of this alternative. Pomerance, Selfridge and Wagstaff [5] show that a strong pseudoprime is always an Euler pseudoprime, but not vice versa, so criterion (2) is indeed stronger than (1). Rotkiewicz [ 10], [ 11 ] proved that every arithmetic progression ax + b (x = 0, 1, 2 . ..) where (a, b) = 1, contains infinitely many ordinary pseudoprimes (that is to say, pseudoprimes for the base 2). Received July 20, 1981. 1980 Mathematics Subject Classification. Primary lOA15; Secondary 10-04. Kev words and phrases. Pseudoprime, Lucas sequence, Lucas pseudoprime, Lehmer numbers, Lehmer sequence, strong pseudoprime, Euler pseudoprime, Euler Lehmer pseudoprime, strong Lehmer pseudoprime, primality testing. ?1982 American Mathematical Society 0025-5718/81 /0000-0444/$03 25

On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with parameters 𝐿, 𝑄 in arithmetic progressions