Revised Mar 10, 2015 @ 21:40. Initially posted Jan 9, 2014 @ 9:04.
I admit to being neither mathematician nor philosopher, nor even a particularly thorough science fiction fan, but I enjoy wandering around the neighborhoods. At random intervals, I may emit odd observations about their intersections while waiting for the light to change.
Way back in the last millennium, I happened upon Robert P. Munafo’s wondrous Notable Properties of Specific Numbers, and was captivated by Clifford Pickover’s superfactorial operator. I knew of functions that grow large very quickly, or thought I did, but this operator is of a different order entirely.
The superfactorial of a value, written as n$, begins by taking the ordinary factorial of the value, n!, then raising it to the n! power. But not just once. The exponentiation repeats in a “power tower” that is n! levels high. The operator’s stupendous growth is evident by the third iteration:
1$ = 1
2$ = 2 ^ 2 = 4
3$ = 6 ^ 6 ^ 6 ^ 6 ^ 6 ^ 6 ≈ 10 ^ 10 ^ 10 ^ 10 ^ 36305
Since the value of 3$ far exceeds a googolplexplexplex (10 ^ 10 ^ 10 ^ 10 ^ 100), it should be obvious that the superfactorial operator is a tool with no practical use whatsoever, except perhaps to (some small subset of) mathematicians and philosophers.
Large Friendly Numbers
Eventually, the idea of applying the superfactorial operator to 42 occurred to me, as it naturally would to even casual fans of Douglas Noel Adams’ The Hitchhiker’s Guide To The Galaxy. As you will recall, 42 is The Answer, calculated over 7-1/2 million years by the computer Deep Thought in the expectation of a meaningful response to the question of Life, the Universe, and Everything. Despite the noticeably quantized and mathematical nature of Nature, there was still some disappointment attending the revelation of 42, but Deep Thought rightly pointed out that more work was needed to refine The Question, before The Answer would make much sense. For our purposes here, we will stick with and expand upon 42. (“Though I don’t think,” added Deep Thought, “that you’re going to like it.”)
A refinement of the concept of 42$ soon followed, almost inevitably it seemed. I wondered if there might be a prime number in the near vicinity of 42$. Actually, in the inaugural version of this silliness, I speculated that simply adding 1 might do the trickiness, but a couple of hoopy froods pointed out that I had my towel wrapped around my head. (Thanks, Lucas and Geoffrey!) Glaringly obvious in retrospect, 1 is exactly the wrong offset to add, as the result matches one of the four patterns for factoring binomial expressions. That’s better than subtracting 1, I suppose, which fails to be prime in two different ways (quadratically and cubically). Replacing the 1 with a prime number bigger than 42 avoids the possibility of any common factors between the two terms, dispensing with the remaining pattern. But what number is compelling enough to recommend itself for the job? A candidate has emerged of late, and that candidate is 421.
When You’re Halving More Than One
The number 421 is not merely a 42 tacked onto the front of a 1, although that makes quite a strong case right off, with its undoubling down progression intrinsic to discrete mathematics, computers, and losing all your money at gambling. As Castañeda might have put it in Tails Of Power, “Cuatro, dos, uno, *nada*! It simply ends … vanishes without a tres.” 421 is prime and is indeed the only number of that pattern that is prime, up through the 23-digit number 10,245,122,561,286,432,168,421. It happens to be the number of days between Douglas Adams’ birth date and mine, a fact which amuses me no end and why I came to consider it. Call it Zeno’s Birthday Paradox.
Having that pesky aspect resolved, I can update my speculation and assert it here anew (still naming it quite immodestly) —
The WEB-DNA Conjecture: 42$+421 is prime
I am pleased to report (although contend may be the more proper term) that this assertion is literally impossible to prove. And by literally, I literally mean literally. It may very well be impossible to disprove, but that’s a somewhat less obvious assertion. If this conjecture is to be proven either way, it will have to be done abstractly and with something of the expressive efficiency of the few characters in 42$+421, or in a fashion like unto the binomial factoring mentioned above, because if it comes to establishing primacy via the ordinary approach of checking its divisibility by all prime numbers up to the square root of the number in question, brute force will fail us.
Let Me Googol That For You
To get the tiniest hint of the enormity of the problem encoded in a mere 7 characters, let’s try expanding 42$+421 into its fully operational deathstar form. The first step is to compute the base factorial:
42! = 42 * 41 * 40 * … * 3 * 2 * 1
That wasn’t so bad, although with 52 digits, 1.4 sexdecillion-something-something is already a very large number. It’s not the sort of number you’d encounter except in Really Big Problems, like estimating what our National Debt will be in 2053, or the long odds that I will win a Nobel Prize for my groundbreaking work in Conjecturing.
For those of you with an appreciation for all things mathematical, I will point out that 42! is a highly composite number, since it includes all of the first 13 primes as factors in non-increasing amounts. It is also practical and abundant. It is 41-smooth and has exactly 258,048,000 divisors. The rest of you may merely wish to avoid talking about abundance and practicality with mathematicians.
Despite sexdecillion‘s appealing name, it’s hard to get a feel for the size of the thing. Perhaps seeing it in more familiar terms might help. Here are a few equivalents:
a million billion billion billion billion billion
a billion billion billion trillion trillion
a million billion trillion trillion trillion
a million billion billion trillion quadrillion
a billion trillion quadrillion quadrillion
a million trillion quadrillion quintillion
The value of 42! is not very big compared with the well-known 101-digit value googol (10^100), although it exceeds the square root of a googol (10^50).
Mostly Harmless Might Just Kill You
Now that we have our base number, we need merely to construct our power tower:
42$ = 42! ^ 42! ^ 42! ^ 42! ^ 42! ^ 42! ^ . . . ^ 42! ^ 42! ^ 42! ^ 42! ^ 42! ^ 42!
≈ 1.4 sexdecillion raised to the 1.4 sexdecillion power for 1.4 sexdecillion levels
Writing all that out in longhand is bound to give you a cramp. Besides the increasingly tedious calculations, the number of digits at successive levels grows at a dismaying pace. Even if the impossible rate of 1 second per level could be maintained, and if each level could be scaled down to fit on your iPapyrus tablet without loss of precision (like with the shrink-ray in the game Scale), that many levels would take 45 tredecillion years or so to complete. That’s roughly a billion-trillion-trillion times the current age of the universe, so you probably won’t finish before you could watch your own frantic, failing efforts from the comfort of Milliways (the Restaurant At The End Of The Universe), if only you hadn’t been too busy to book your reservation well ahead of time.
Your Hyperspace Bypass Surgery
Obviously, we shall require some assistance, some absurdly capable computer to swiftly run through these calculations, add the 421, and display our conjectural WEB-DNA value, so we can begin checking whether it’s prime. But we have a problem with this Challenge of Really Astronomical Proportions. This is going to be a very, very, very big number. We may encounter some difficulty simply finding space to store it all.
Deep Thought wasn’t quite up to the task it was given, and eventually turned the work over to a planet-sized simulation that it had designed to run for an even longer period of time. (“And it shall be called … the Earth.” Phouchg gaped at Deep Thought. “What a dull name,” he said.) However, we need something way bigger. And smaller.
In fact, let’s dispense with trying to use ordinary matter for our scratchpad, because with a mere 10 ^ 80 atoms in the universe, there simply aren’t anywhere near enough to get us very far. Perhaps we can make do with pure bits of information, which seems like a splendid Total Perspective Vortex idea. Data just wants to be free, it has been said, but somebody still has to pay its rent. Forty years ago it was proved that a bit of information falling into a black hole increases its event horizon by a Planck area, which is the Planck length squared. I believe that’s more or less the same as saying a single bit will fill a Planck volume. Don’t fret over any of this. The nice thing about the Planck volume, the part that matters in this context, is that it’s really tiny. You could jam roughly 10 ^ 60 of these Planck volumes into a single proton, if you’ve got a steady hand and a great deal of patience.
William Gibson once observed, “The future is already here; it’s just not evenly distributed.” In fact, the entire universe, besides its annoying skill at hiding from us, is terribly lumpy. And some of its larger clumps are black holes, which are worse than the memory holes in either 1984 or BIOS. Calling space-time a continuum seems overstated, in much the same way that it’s hard to argue for any abundant practicality of Hadrian’s Wall as a defense against marauding Scots philosophers from the star fields on the other side of the wall, what with much of it gone missing. Nonetheless, if we could completely fill the entire visible universe with our tiny Planck volume information storage units, and ignoring the questions of how to address and access them all, we could corral about 8.5 × 10 ^ 185 bits. Raising 2 to that number gives us a maximum value that is much larger than a googolplex, about 10 ^ 10 ^ 185.
Yay! Or, rather, Boo! Alas, it’s not even a scratch on the value of 3$ that we saw earlier. It’s much larger than the value of the top 2 layers of our towering little inferno, but completely out of the running by the 3rd level.
Top level of 42$ ≈ 10 ^ 51
Top 2 levels of 42$ ≈ 10 ^ 10 ^ 53
Our imaginary Planck volume notepad ≈ 10 ^ 10 ^ 185
Top 3 levels of 42$ ≈ 10 ^ 10 ^ 72 sexdecillion
Apparently, we will never come anywhere near writing down the value that 42$+421 represents. Merely recording the trailing 0 bits from, say, the first 42 levels of the tower would exceed the size of Reality in all its super-holographic pandimensionality, no matter how much Superstring we try to unwrap it in. It’s like the Vogon Constructor Fleet showed up before we could even discover how to grow hops, much less get a decent pint pulled.
The World Is Not Enough Quantum Of Solace
The Universe As Tabulator gets us through the first 2 levels of our tower. Only 1,405,006,117,752,879,898,543,142,606,244,511,569,936,383,999,999,998 more levels of exponentiation to go! And we haven’t even begun to think about how to perform calculations on our unwieldy number to determine whether it has any factors, which would take many, many more multiverse-lifetimes. To make matters worse, solving a factoring problem like this is much more difficult than the garden-variety Ultimate Question sort, because we cannot use logarithms and other space-, time-, and sanity-saving shortcuts. We need the full precision of every blesséd bit of the entire integer, in the correct order. And it would be a magical universe indeed that suffered no random errors in writing, storing, and reading bits.
On the plus side, there are some things we can know about 42$+421. Because it is a single 3-digit prime number greater than many hideously huge multiples of the first 13 prime numbers, it cannot be divisible by any of them: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, and/or 421. And we know how the number ends. The last 17.5% or so of its digits consist of 6 and 421 separated by an uncountably long string of zeros.
But carry on the calculations until we are all carried off, we could never figure out how it begins.
All The World’s A Stage We’re Passing Through
It was once speculated that given a sufficiently long interval, a sufficiently large pool of monkeys banging on a collection of sufficiently working keyboards would recreate literary works of a sufficiently Shakespearean nature, assuming you had the patience to wade through all the inane social media-like gibberish that would necessarily dominate the output. We have in this Forty-Two-Superfactorial Number sublime echoes of the romance and mystique derived from its antecedent 42, The Answer to Life, The Universe, and Everything. Improbably, the literary works of Douglas Adams may well appear somewhere in the vast middle of its Larger-than-Life-The-Universe-and-Everything bit sequence. Maybe backwards. Likely in Vogon.
This Number is conceptually a plain old, ordinary, finite integer. But its unwieldy size opens a philosophical can of skeptical worms. What does finite mean at this range? Is it rational to claim that a number that cannot in any sense fit within any conceivably ordinary space-time dimensions is really finite?
Richard Feynman once remarked, “It doesn’t seem to me that this fantastically marvelous universe … can merely be a stage so that God can watch human beings struggle for good and evil — which is the view that religion has. The stage is too big for the drama.”
Perhaps. Yet, we have a straightforward question — Is 42$+421 a prime number? — that fits perfectly well within ordinary if slightly warped minds, for which the stage is nonetheless entirely too small. And it’s nowhere near the largest question that can be imagined.
There is, for example, whether (42$)$ + 421 is prime.
We apologize for the inconvenience.