Prime Numbers

3.9 Research problems 169
with W (n) =k, letnk denote the smallest such n. Wehave
n2 = 9 n12 > 10 16
n3 = 2047 n13 = 2152302898747
n5 = 1373653 n14 = 1478868544880821
n6 = 134670080641 n17 = 3474749660383
n7 = 25326001 n19 = 4498414682539051
n10 = 307768373641 n23 = 341550071728321.
n11 = 3215031751
(These values were computed by D. Bleichenbacher, also see [Jaeschke 1993],
[Zhang and Tang 2003], and Exercise 4.34.) S. Li has shown that W (n) =12
for
n = 1502401849747176241,
so we know that n12 exists. Find n12 and extend the above table. Using
Bleichenbacher’s computations, we know that any other value of nk that exists
must exceed 10 16 .
3.44. Study, as a possible alternative to the simple trial-division Algorithm
3.1.1, the notion of taking (perhaps extravagant) gcd operations with the N
to be factored. For example, you could compute a factorial of some B and
take gcd(B!,N), hoping for a factor. Describe how to make such an algorithm
complete, with the full prime factorizations resulting. This completion task is
nontrivial: For example, one must take note that a factor k 2 of N with k