I completed the project (as complete as it will ever be). I ended up using one child thread with a queue to create two parallel processes. It was exactly what I wanted, and it was extremely simple to implement. The main process created the valid numbers; the child process paired them up. The two ran very much in parallel since the pairing algorithm actually ran more efficiently working on a growing list of numbers. If I had more processors, I'm not sure how I'd partition the logic.

I think I'd have to re-frame the overall algorithm to take advantage of many processors; then, it may have been more efficient to multi-thread and partition like (some of) you suggested.

The time dropped to 44% of the original (2.25 times faster). Some of the speed came not from the threading, but just learning how to write more efficient perl statements. I learned that there are expensive operations in perl and some very fast operations... I ended up tweaking many, and changing how I was storing the data to take advantage of these realities.

I am sure the program could be made even faster once I understand perl a bit better.

Here are the code snippets relevant to the threading:

# This child thread builds an encoded array of valid pairs.
#  "valid pairs" do not have any of the same bits set.
# Output format is row1_integer:row2_integer
my $DataQueue = Thread::Queue->new();
my ($thr) = threads->create({'context' => 'list'}, sub
{
    # I tried to use a shared hash of hashes, but the required work to
+ explicity define the hash of hashes in the manner required by thread
+s::shared was just too expensive.
    use integer;
    # This thread uses queues to get its parameter.
    my @ValidPairs=();
    my @output=();
    while (my $validNum = $DataQueue->dequeue())
    {
        !($validNum & $_) and push @output, $validNum .':'.$_ foreach 
+(@ValidPairs);
        push (@ValidPairs, $validNum ); #must add number to this threa
+d's list of valid numbers
    }
    return (@output);
});

#... Inside loop and logic to build numbers
      # send this new number to the child thread waiting on queued par
+ameters.
      # The child thread builds the array of valid pairs 
      $DataQueue->enqueue($i);
#... After block that builds the numbers
$DataQueue->enqueue(undef); #tell the child process to finish what has
+ been queued and then exit.
my @encodedPairedNumbers = $thr->join();

#subsequent code decodes our encoded paired numbers into a hash of has
+hes
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Thank you to everyone for your help!