Prime Numbers

164 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
factor exceeds n c (when there are less than two prime factors let us simply
not count that n). For the high-precision c value given above, there are 548
such n ∈ [1, 1000], whereas the theory predicts 500. Give the count for some
much higher value of x.
3.6. Rewrite the basic Eratosthenes sieve Algorithm 3.2.1 with improvements.
For example, reduce memory requirements (and increase speed) by
observing that any prime p>3 satisfies p ± 1 (mod 6); or use a modulus
greater than 6 in this fashion.
3.7. Use the Korselt criterion, Theorem 3.4.6, to find by hand or machine
some explicit Carmichael numbers.
3.8. Prove that every composite Fermat number Fn = 22n+1 isa Fermat pseudoprime base 2. Can a composite Fermat number be a Fermat
pseudoprime base 3? (The authors know of no example, nor do they know a
proof that this cannot occur.)
3.9. This exercise is an exploration of rough mental estimates pertaining
to the statistics attendant on certain pseudoprime calculations. The great
computationalist/theorist team of D. Lehmer and spouse E. Lehmer together
pioneered in the mid-20th century the notion of primality tests (and a great
many other things) via hand-workable calculating machinery. For example,
they proved the primality of such numbers as the repunit (10 23 − 1)/9 with
a mechanical calculator at home, they once explained, working a little every
day over many months. They would trade off doing the dishes vs. working on
the primality crunching. Later, of course, the Lehmers were able to handle
much larger numbers via electronic computing machinery.
Now, the exercise is, comment on the statistics inherent in D. Lehmer’s
(1969) answer to a student’s question, “Professor Lehmer, have you in all
your lifetime researches into primes ever been tripped up by a pseudoprime
you had thought was prime (a composite that passed the base-2 Fermat
test)?” to which Lehmer’s response was as terse as can be: “Just once.” So
the question is, does “just once” make statistical sense? How dense are the
base-2 pseudoprimes in the region of 10 n ? Presumably, too, one would not
be fooled, say, by those base-2 pseudoprimes that are divisible by 3, so revise
the question to those base-2 pseudoprimes not divisible by any “small” prime
factors. A reference on this kind of question is [Damg˚ard et al. 1993].
3.10. Note that applying the formula in the proof of Theorem 3.4.4 with
a = 2, the first legal choice for p is 5, and as noted, the formula in the proof
gives n = 341, the first pseudoprime base 2. Applying it with a = 3, the first
legal choice for p is 3, and the formula gives n = 91, the first pseudoprime
base 3. Show that this pattern breaks down for larger values of a and, in fact,
never holds again.
3.11. Show that if n is a Carmichael number, then n is odd and has at least
three prime factors.