Lucas Pseudoprimes

Abstract

Abstract. We define several types of pseudoprimes with respect to Lucas sequences and prove the analogs of various theorems about ordinary pseudoprimes.For example, we show that Lucas pseudoprimes are rare and we count the Lucas sequences modulo n with respect to which n is a Lucas pseudoprime. We suggest some powerful new primality tests which combine Lucas pseudoprimes with ordinary pseudoprimes.Since these tests require the evaluation of the least number f(n) for which the Jacobi symbol (f(n)/n) is less than 1, we evaluate the average order of the function /.

The pseudoprimes to 25⋅10⁹

“…By the above, 2" -1 is an spsp (2). Since it is divisible by 2P -1 for each divisor p of«, and since the numbers 2P -1 with distinct primes p are relatively prime, 2" -1 has at least as many prime factors as «.…”

“…Thus, the strong form of the prime fc-tuples conjecture implies that at least cx1'2/ln2x integers below x are simultaneously psp (2) and psp(3). Table 6 Number of pseudoprimes to bases 2, 3, 5, 7 below a limit 9.…”

“…If n is a psp(2), then 2" -1 is an spsp (2). There exist spsp(2)'s with arbitrarily many prime divisors.…”

“…Table 5 gives the number of psp(2), spsp(2), and Carmichaels below 25 • IO9 which have exactly k prime factors (counted according to multiplicity). Observe that the spsp(2)'s usually have only two prime factors and that the Carmichaels have more prime factors than the typical psp (2). Of course, the Carmichaels must have at least three prime factors, but the discrepancy is more than can be so explained.…”

Higher-order Carmichael numbers

“…Clearly one can generalize the idea of a Carmichael number by allowing the pseudoprimality test in the definition to vary over some larger class of tests (perhaps including some of those found in [1], [2], [4], [6], [8], [9], [11], [16], [19], [26]), and indeed such generalizations have been considered (see for example [5], [8], [13], [15], [17], [18], [19], [21], [22], [27]). But there is also a natural algebraic way of generalizing the concept of a Carmichael number that makes no mention of pseudoprimality.…”