On Carmichael Numbers with 3 Factors and the Strong Pseudoprime Test an Odd Composite Integer N Is a \carmichael Number" If a N a Mod N for All Integers A. It Is a \3-factor
نویسنده
چکیده
| The well known \strong pseudoprime test" has its highest probability of error (1=4) when the numbers being tested are certain Carmichael numbers with 3 prime factors. We present a non-rigorous plausibility argument that the count C3 (x) of 3-factor Carmichaels up to x, is asymptotically C3(x) K x 1=3 (ln x) ?3 where K 478 110 is an absolute constant given by K = 3 3 X 1a 1=8 in the strong pseudoprime test are certain numbers with only 2 prime factors and certain prime powers. However if these cases are somehow known to apply, then we show how to improve the strong pseudoprime test so that its probability of error on N is O(1= p ln N). Carmichael" if it has exactly 3 prime factors. ajb means \a divides b" and a 6 jb means \a does not divide b." In expressions P p or Q p , we mean for p to range over the
منابع مشابه
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