# A Number Theory Puzzle Involving Primes, Perfect Numbers, and “Paraperfect” Numbers

There is something about the definitions of prime and perfect numbers that always struck me as rather odd. Prime numbers are those which no factors other than themselves, and one. Perfect numbers, on the other hand, equal the sum of all of their factors, excluding themselves, but including one. The first two examples of perfect numbers are 6 (which equals 1 + 2 + 3) and 28 (which equals 1 + 2 + 4 + 7 + 14). Perfect numbers are far more rare than primes.

The thing I find annoying is the exclusion of one, as a factor, from one of these definitions, but not the other. I therefore decided to give a name to a new type of number:  one which equals the sum of its factors, excluding itself AND the number one. The first name I thought of, “exceptional numbers,” turns out to have already been taken, so I thought of another, and called these numbers “paraperfect numbers” instead.

Having done that, it was time to start searching for them. I have a reasonably fast mental calculator, but it didn’t take long to figure out that I wasn’t up to this task, so I wrote this program to search for paraperfect numbers:

It’s written in BASIC, an archaic computer language I learned in high school, and, as you can see, I am a horribly sloppy programmer. A better programmer would have written a program for the same purpose, but with only about half this length. Be that as it may, though, the program does work. I’ve had it running for a few minutes now.

It’s gotten past 22,000 — and has found no paraperfect numbers at all. This is not what I expected . . . and now I am wondering if any exist. Right now, of course, “no paraperfect numbers exist” is a mere conjecture. If I can prove it, it will be a theorem. However, I don’t know nearly enough about number theory to write such a proof.

I could use some help. If anyone does find a paraperfect number, please leave a comment on this post identifying your find. If anyone can prove — or simply explain to me — why there are no paraperfect numbers, if that is the case, please let me know that as well.

The program is still running, and has now passed 25,000 without a paraperfect-number find. I guess I’ll leave it running for a while. Any help with this puzzle would be much appreciated.

[Later:  see the comments for the rest of the story on these elusive numbers.]

# My Favorite Passage from the Bible, and How One Atheist Thinks We Just Might Use It to Avoid Extinction.

You may already know I am an atheist, and may be unaware that some of us have favorite passages from the Bible which were not selected for purposes of ridicule, nor of criticism of the Bible, nor because of dislike of any religion. This is my favorite passage because it contains excellent advice. I do not need “faith,” as that word is commonly understood, nor a literal belief in the devil, to recognize, and appreciate, good advice.

What’s not to like about self-control? Or being alert? Those things can keep us all alive. They are important. I used to only cite the first sentence here as my favorite part of the Bible, but have decided to include two complete verses, for context, and elaboration through metaphor, as I interpret this passage. I see no reason not to.

Atheists (only capitalized at the beginning of a sentence, by the way) don’t have denominations, nor creeds, and there are as many different types of atheist as there are atheists. Atheism isn’t a religion — the word simply describes existence without religion. Everyone is born an atheist, albeit an unconscious one. Also, those who remain, or return to, atheism, change, during the course of our lives, just as theists do. The only people who do not change are the dead.

In defiance of stereotype, we are not all angry and bitter, although some of us, it must be admitted, are. (I used to be far more bitter than I am now, although I am working hard to change that.) Many of us even believe in non-theistic ideas which make absolutely no sense, such as, for example, 9/11 conspiracy theories. We only have one thing in common: we lack belief in deities. You almost certainly lack belief in at least some deities, ones which others fervently believe in. If you are a theist, well, atheists just take things a bit further than you — that’s all. We don’t all hate theists, and (thankfully) not all theists hate us. The ability to respectfully disagree is at least one of the keys to peaceful coexistence. Universal agreement among humans simply will not happen (and would be horribly boring, anyway), until the death of the penultimate person, at least. Even if there is a “last person alive” scenario in the (hopefully very distant) future, this unknown last human being will still have internal disagreements, and will almost certainly disagree with remembered ideas of the dead. In fact, given human nature, and history, such a disagreement might even be the cause of the next-to-last person’s death, at the hands of the last man, or woman, ever to live.

I do not want homo sapiens to end this way.  I’d like us to continue, for many generations, until evolution, and speciation, replace us with successor species, a long time from now — still people, but different, in ways we cannot now know, and, hopefully, people who have long ago learned to live without constantly killing each other.  Isn’t it about time we left this nasty habit called “war” behind, along with murder, rape, and the rest of the litany of human horror?

I’m a big fan of John Lennon, but I’d far rather imagine no war than “imagine no religion,” and I no longer accept the idea, common among atheists, that the second is a prerequisite for the first.

Since we have, as a species, figured out several ways to self-destruct, we cannot afford to wait for evolution to “teach” us how to coexist peacefully.  Evolution is far more efficient at destruction than creation, after all, being a random process.  Far more species have gone extinct than exist today, and the process of evolution simply does not care whether we live or die.  Entropy happens.  It took 3.85 billion years of natural selection to get here, and we will not get a second chance to get it right.

We must figure out effective ways to live with our differences now.  I do not mean that we should somehow erase our differences, for I have no desire to live in a world of clones of myself, and I doubt you want to live in your version of such a world, either.  We do, however, need to come to terms, as a world-wide society, with the inescapable fact that people are different.  We have a right to be different, it’s good that we are, and the fact that we vary so much is certainly is no excuse for killing, nor even hating, anyone.

There is another part of human nature that is on our side in our struggle for survival, and this is the hopeful part of this essay. We are good at figuring things out. We actually enjoy trying our best to solve puzzles. We pay hard-earned money for them constantly! Some of us absolutely obsess over single problems, for days — or years — at a time. Well, this is the best, most important problem we have ever faced, with the highest stakes imaginable:  how to avoid our own extinction. The world isn’t a casino with no exit, though.  It has been mostly a game of chance, so far — and we’ve been lucky to have made it to the present.  However, it doesn’t have to be the way it has been, with us stumbling through history, like drunk monkeys in a minefield — which we pretty much are, right now.

We have minds, and it’s time to use them. We can stop playing roulette, especially the Russian variety, and sit down at the table to play chess, instead. We can figure this out.

If this Big Problem isn’t solved soon, though, there may not be a long wait for extinction.  It could very well be later than you think.  Therefore, I encourage everyone to, in the words of the Bible, “Be self-controlled and alert.” That’s a good place to start.