Prime Numbers

4.6 Exercises 219
what calendar year could F33 be resolved via the Pepin test? Note, in this
connection, the itemized remarks pursuant to Table 1.3.
Analyze and discuss these issues:
(1) The possibility of parallelizing the Pepin squaring (nobody knows how to
parallelize the squaring chain overall in an efficient manner, but indeed one
can parallelize within one squaring operation by establishing each element
of a convolution by way of parallel machinery and the CRT).
(2) The problem of proving the character of Fn is what the final Pepin residue
says it is. This is an issue because, of course, a machine can sustain
either cosmic-ray glitches (hardware) or bugs (software) that ruin the
proof. Incidentally, hardware glitches do happen; after all, any computing
machine, physics tells us, lives in an entropy bath; error probabilities are
patently nonzero. As for checking software bugs, it is important to have
different code on different machines that are supposed to be checking each
other—one does not even want the same programmer responsible for all
machines!
On this latter issue, consider the “wavefront” method, in which one, fastest
available machine performs Pepin squaring, this continual squaring thought of
as a wavefront, with other computations lagging behind in the following way.
Using the wavefront machine’s already deposited Pepin residues, a collection
of (slower, say) machines verify the results of Pepin squarings at various
intermediate junctures along the full Pepin squaring chain. For example,
the fast, wavefront machine might deposit the millionth, two millionth, three
millionth, and four millionth squares of 3; i.e., deposit powers
3 21000000
, 3 22000000
, 3 23000000
, 3 24000000
all modulo Fn, and each of the slow machines would grab a unique one of
these residues, square it just one million times, and expect to find precisely
the deterministic result (the next deposited power).
4.7. Prove the following theorems of Suyama (see [Williams 1998]):
(1) Suppose k is an odd number and N = k2 n + 1 divides the Fermat number
Fm. Prove that if N