“Yo no tomo drogas. Yo soy una droga.” — Salvador Dalí (1904-89).
“I do not take drugs. I am a drug.”
Author Archives: Krilling for Company
In Perms Of
13 is a prime number, divisible only by itself and 1. Perm 13 and you get 31, which is also a prime number. The same is true of 17, 37 and 79. There are only two possible permutations – 2 x 1 – of a two-digit number, so base-10 is terminally permal for two-digit primes:
13, 31 17, 71 37, 73 79, 97
What about three-digit primes? Now there are six possible permutations: 3 x 2 x 1. But base-10 is not terminally permal for three-digit primes. This is the best it does:
149, 419, 491, 941 179, 197, 719, 971 379, 397, 739, 937
Fortunately, we aren’t restricted to base-10. Take a step up and you’ll find that base-11 is terminally permal for three-digit primes (139 in base-11 = 1 x 11^2 + 3 x 11 + 9 = 163 in base-10):
139, 193, 319, 391, 913, 931 (6 primes) (base=11) 163, 223, 383, 463, 1103, 1123 (base=10)
Four-digit primes have twenty-four possible permutations – 4 x 3 x 2 x 1 – and base-10 again falls short:
1237, 1327, 1723, 2137, 2371, 2713, 2731, 3217, 3271, 7213, 7321 (11 primes) 1279, 1297, 2179, 2719, 2791, 2917, 2971, 7129, 7219, 9127, 9721
For four-digit primes, the most permal base I’ve discovered so far is base-13 (where B represents [11]):
134B, 13B4, 14B3, 1B34, 1B43, 314B, 31B4, 34B1, 3B14, 413B, 41B3, 431B, 43B1, 4B13, 4B31, B134, B143, B314, B413 (19 primes) (base=13) 2767, 2851, 3019, 4099, 4111, 6823, 6907, 7411, 8467, 9007, 9103, 9319, 9439, 10663, 10687, 24379, 24391, 24691, 24859 (base=10)
Is there a base in which all permutations of some four-digit number are prime? I think so, but I haven’t found it yet. Is there always some base, b, in which all permutations of some d-digit number are prime? Is there an infinity of bases in which all permutations of some d-digit number are prime? Easy to ask, difficult to answer. For me, anyway.
Ass You Like It
This is a guest post by Norman Foreman, B.A.
Mediaeval Catholic philosophers wrote about both praying and braying. The braying came from Buridan’s ass, a thought-experiment about choice and free will. Imagine a hungry ass set between two piles of hay that are identical in every way: size, shape, colour, tastiness and so on. Some philosophers argued that, if it had no reason to prefer one pile of hay to the other, the ass would be unable to choose and would therefore starve to death.
I don’t agree: inter alia, nervous systems don’t work symmetrically and we don’t experience objects as fully identical when they’re in different parts of our visual field. However, in a literary sense, I understand what it feels like to be Buridan’s ass. To assify myself, I start by imagining this:
• I’m offered £1000 to read a book by the transgressive author Will Self.
Would I accept? Yes. It would be distasteful, but I’d do it for £1000. Self’s writing is so bad that I might give the money back rather than finish the book, but I’d have a go. Now change the situation:
• I’m offered £1000 to read a book by the transgressive author Stewart Home.
Would I still accept? Yes. Again, it would be distasteful, but I’d do it for the money. Or I’d try, at least. The next step turns me into Buridan’s ass. I imagine this:
• I’m offered £1000 to read a book by either Will Self or Stewart Home (not both). And I have to make the choice for myself.
Now I’m on the horns of a dilemma. I would want the £1000, but I can’t decide which transgressive author I’d rather NOT read. Home is a downmarket version of Self, Self is an upmarket version of Home. It’s Self-as-chav vs Home-as-Oxbridge-grad. And/or vice versâ. They’re both keyly committed core components of the Guardianista community, with all that that implies in terms of issues around bad English, mixed metaphors and “in terms of”. I’m happy to say I’ve never read a book by either of them. So if I were offered £1000 to do so and had to choose either Self or Home, I couldn’t do it. Not unassisted. I’d have to toss a coin. Best of three. Or best of five dot dot dot
Previously pre-posted (please peruse):
Performativizing Papyrocentricity #12
Papyrocentric Performativity Presents:
Hawt’ in the Act – Whatshisname: The Life and Death of Charles Hawtrey, Wes Butters (Tomahawk Press 2010)
Lez Redd – The Trials and Triumphs of Les Dawson, Louis Barfe (Atlantic Books 2012)
Fetch and Carry – The Surfrider, compiled by Jack Pollard (K.G. Murray 1963)
Bri’ on the Sky – Wonders of the Solar System, Professor Brian Cox and Andrew Cohen (Collins 2010) (posted @ Overlord of the Über-Feral)
Playing on the Nerves – In A Glass Darkly, Sheridan Le Fanu (@ O.o.t.Ü.-F.)
Or Read a Review at Random: RaRaR
Bri’ on the Sky
Bri’ Eyes the Sky
One of the most powerful images in this book is also one of the most understated. It’s an artist’s impression of a dim star seen over the curve of a dwarf-planet called Sedna. The star is a G-type called Sol. We on Earth know it better as the sun. Sedna is a satellite of the sun too, but it’s much, much further out than we are. It takes 12,000 years to complete a single orbit and its surface is a biophobic -240°C. It’s so distant that sunrise is star-rise and it wasn’t discovered until 2003. But the sun’s gravity still keeps it in place: one of the weakest forces in nature is one of the most influential. That’s one important message in an understated, crypto-Lovecraftian image.
Sedna has been there, creeping around its dim mother-star, since long before man evolved. It will still be there long after man disappears, voluntarily or otherwise. This frozen dwarf is a good symbol of the vastness of the universe and its apparent indifference to life. We don’t seem to interest the universe at all, but the universe certainly interests us. Wonders of the Solar System is a good introduction to our tiny corner of it, describing some fundamentals of astronomy with the help of spectacular photographs and well-designed illustrations. You can learn how fusion powers the sun, how Mars lost its atmosphere and how there might be life beneath the frozen surface of Jupiter’s satellite Europa. The text is simple, but not simplistic, though I think the big name on the cover did little of the writing: this book is probably much more Cohen than Cox. Either way, I enjoyed reading the words and not just looking at the pictures, all the way from star-dim Sedna (pp. 26-7) to “Scars on Mars” (pp. 220-1) by way of “The most violent place in the solar system” (pp. 198-9), a.k.a. Jupiter’s gravity-flexed, volcano-pocked satellite Io.
Pockmarked moon — the Galilean satellite Io
Everything described out there is linked to something down here, because that’s how it was done in the television series. Linking the sky with the earth allowed the BBC to film the genial and photogenic physicist Brian Cox in various exotic settings: Hawaii, India, East Africa, Iceland and so on. I’ve not seen any of Cox’s TV-work, but he seems an effective popularizer of science. And the pretty-boy shots here add anthropology to the astronomy. What is the scientific point of Cox striding away in an artistic blur over the Sahara desert (pg. 103), staring soulfully into the distance near the Iguaçu Falls on the Brazilian-Argentine border (pg. 37) or gazing down into the Grand Canyon, hips slung, hands in pockets (pg. 163)? There isn’t a scientific point: the photos are there for his fans, particularly his female ones. He’s a sci-celeb, a geek with chic, and we’re supposed to see the sky through Bri’s eyes.
But he’s also a liberal working for the Bolshevik Broadcasting Corporation, so he’ll be happy with the prominent photo early on: Brian holding protective glasses over the eyes of a dusky-skinned child during a solar eclipse in India. The same simul-scribes’ Wonders of Life (Collins 2013), another book-of-the-BBC-series, opens with a similarly allophilic allophoto: a dusky-skinned Mexican crowned in monarch butterflies. This is narcissistic and patronizing, but the readiness of whites to “Embrace the Other” helps explain science, because science involves looking away from the self, the tribe and the quotidian quest for status and survival. Of course, Cox and Cohen would gasp with horror at the idea of racial differences explaining big things like science and politics. Cox would be sincere in his horror. I’m not so sure about Cohen.
But there are wonders within us as well as without us and though you won’t hear about them on the BBC, the tsunami of HBD, or research into human bio-diversity, is now rolling ashore. It will sweep away almost all of Cox’s and Cohen’s politics, but leave most of their science intact. It isn’t a coincidence that the rings of Saturn were discovered by the Italian Galileo and explained by the Dutchman Huygens and the Italian Cassini, or that the photos of Saturn here were taken by a space-probe launched by white Americans. But the United States has much less money now for space exploration. That’s explained by race too: as the US looks less like its founders, it looks less like a First World nation too. It’s fun to see the world through Bri’s eyes, but he’s careful not to look at everything that’s out there.
Factory Records
The factors of n are those numbers that divide n without remainder. So the factors of 6 are 1, 2, 3 and 6. If the function s(n) is defined as “the sum of the factors of n, excluding n”, then s(6) = 1 + 2 + 3 = 6. This makes 6 a perfect number: its factors re-create it. 28 is another perfect number. The factors of 28 are 1, 2, 4, 7, 14 and 28, so s(28) = 1 + 2 + 4 + 7 + 14 = 28. Other perfect numbers are 496 and 8128. And they’re perfect in any base.
Amicable numbers are amicable in any base too. The factors of an amicable number sum to a second number whose factors sum to the first number. So s(220) = 284, s(284) = 220. That pair may have been known to Pythagoras (c.570-c.495 BC), but s(1184) = 1210, s(1210) = 1184 was discovered by an Italian schoolboy called Nicolò Paganini in 1866. There are also sociable chains, in which s(n), s(s(n)), s(s(s(n))) create a chain of numbers that leads back to n, like this:
12496 → 14288 → 15472 → 14536 → 14264 → 12496 (c=5)
Or this:
14316 → 19116 → 31704 → 47616 → 83328 → 177792 → 295488 → 629072 → 589786 → 294896 → 358336 → 418904 → 366556 → 274924 → 275444 → 243760 → 376736 → 381028 → 285778 → 152990 → 122410 → 97946 → 48976 → 45946 → 22976 → 22744 → 19916 → 17716 → 14316 (c=28)
Those sociable chains were discovered (and christened) in 1918 by the Belgian mathematician Paul Poulet (1887-1946). Other factor-sum patterns are dependant on the base they’re expressed in. For example, s(333) = 161. So both n and s(n) are palindromes in base-10. Here are more examples — the numbers in brackets are the prime factors of n and s(n):
333 (3^2, 37) → 161 (7, 23)
646 (2, 17, 19) → 434 (2, 7, 31)
656 (2^4, 41) → 646 (2, 17, 19)
979 (11, 89) → 101 (prime)
1001 (7, 11, 13) → 343 (7^3)
3553 (11, 17, 19) → 767 (13, 59)
10801 (7, 1543) → 1551 (3, 11, 47)
11111 (41, 271) → 313 (prime)
18581 (17, 1093) → 1111 (11, 101)
31713 (3, 11, 31^2) → 15951 (3, 13, 409)
34943 (83, 421) → 505 (5, 101)
48484 (2^2, 17, 23, 31) → 48284 (2^2, 12071)
57375 (3^3, 5^3, 17) → 54945 (3^3, 5, 11, 37)
95259 (3, 113, 281) → 33333 (3, 41, 271)
99099 (3^2, 7, 11^2, 13) → 94549 (7, 13, 1039)
158851 (7, 11, 2063) → 39293 (prime)
262262 (2, 7, 11, 13, 131) → 269962 (2, 7, 11, 1753)
569965 (5, 11, 43, 241) → 196691 (11, 17881)
1173711 (3, 7, 11, 5081) → 777777 (3, 7^2, 11, 13, 37)
Note how s(656) = 646 and s(646) = 434. There’s an even longer sequence in base-495:
33 → 55 → 77 → 99 → [17][17] → [19][19] → [21][21] → [43][43] → [45][45] → [111][111] → [193][193] → [195][195] → [477][477] (b=495) (c=13)
1488 (2^4, 3, 31) → 2480 (2^4, 5, 31) → 3472 (2^4, 7, 31) → 4464 (2^4, 3^2, 31) → 8432 (2^4, 17, 31) → 9424 (2^4, 19, 31) → 10416 (2^4, 3, 7, 31) → 21328 (2^4, 31, 43) → 22320 (2^4, 3^2, 5, 31) → 55056 (2^4, 3, 31, 37) → 95728 (2^4, 31, 193) → 96720 (2^4, 3, 5, 13, 31) → 236592 (2^4, 3^2, 31, 53)
I also tried looking for n whose s(n) mirrors n. But they’re hard to find in base-10. The first example is this:
498906 (2, 3^3, 9239) → 609894 (2, 3^2, 31, 1093)
498906 mirrors 609894, because the digits of each run in reverse to the digits of the other. Base-9 does better for mirror-sums, clocking up four in the same range of integers:
42 → 24 (base=9)
38 (2, 19) → 22 (2, 11)
402 → 204 (base=9)
326 (2, 163) → 166 (2, 83)
4002 → 2004 (base=9)
2918 (2, 1459) → 1462 (2, 17, 43)
5544 → 4455 (base=9)
4090 (2, 5, 409) → 3290 (2, 5, 7, 47)
Base-11 does better still, clocking up eight in the same range:
42 → 24 (base=11)
46 (2, 23) → 26 (2, 13)
2927 → 7292 (base=11)
3780 (2^2, 3^3, 5, 7) → 9660 (2^2, 3, 5, 7, 23)
4002 → 2004 (base=11)
5326 (2, 2663) → 2666 (2, 31, 43)
13772 → 27731 (base=11)
19560 (2^3, 3, 5, 163) → 39480 (2^3, 3, 5, 7, 47)
4[10]7[10]9 → 9[10]7[10]4 (base=11)
72840 (2^3, 3, 5, 607) → 146040 (2^3, 3, 5, 1217)
6929[10] → [10]9296 (base=11)
100176 (2^4, 3, 2087) → 158736 (2^4, 3, 3307)
171623 → 326171 (base=11)
265620 (2^2, 3, 5, 19, 233) → 520620 (2^2, 3, 5, 8677)
263702 → 207362 (base=11)
414790 (2, 5, 41479) → 331850 (2, 5^2, 6637)
Note that 42 mirrors its factor-sum in both base-9 and base-11. But s(42) = 24 in infinitely many bases, because when 42 = 2 x prime, s(42) = 1 + 2 + prime. So (prime-1) / 2 will give the base in which 24 = s(42). For example, 2 x 11 = 22 and 22 = 42 in base (11-1) / 2 or base-5. So s(42) = 1 + 2 + 11 = 14 = 2 x 5 + 4 = 24[b=5]. There are infinitely many primes, so infinitely many bases in which s(42) = 24.
Base-10 does better for mirror-sums when s(n) is re-defined to include n itself. So s(69) = 1 + 3 + 23 + 69 = 96. Here are the first examples of all-factor mirror-sums in base-10:
69 (3, 23) → 96 (2^5, 3)
276 (2^2, 3, 23) → 672 (2^5, 3, 7)
639 (3^2, 71) → 936 (2^3, 3^2, 13)
2556 (2^2, 3^2, 71) → 6552 (2^3, 3^2, 7, 13)
In the same range, base-9 now produces one mirror-sum, 13 → 31 = 12 (2^2, 3) → 28 (2^2, 7). Base-11 produces no mirror-sums in the same range. Base behaviour is eccentric, but that’s what makes it interesting.
I Say, I Sigh, I Sow #9
Playing on the Nerves
In A Glass Darkly, Sheridan Le Fanu
Far less known than his great admirer M.R. James, the Dubliner Sheridan Le Fanu (1814-73) may be an even better and more haunting writer. And yet he doesn’t rely much on the supernatural. Some of his stories seem to be more about neurological disease than about ghostly visitation. That kind of disease was much more common in his Georgian and Victorian day, when the toxicity of many chemicals wasn’t understood properly and people could be poisoned by arsenic in their wallpaper. But the horrors conjured by a diseased brain can be both stronger and more mysterious than a ghost or demon, because they’re more intimate and less easy to escape.
Le Fanu is intimate in another way: he has Robert Aickman’s ability to start currents swirling in your subconscious. You can feel yourself being drawn down into the abysses that wait there, dark and mysterious with sex, death and primal instinct. “Carmilla”, his classic tale of adolescent lesbian vampirism, is a good example. It also reveals his wider sympathy with humanity. M.R. James would not have written about women or about that kind of sex. Homosexuality and necrophilia seem to inform James’ stories; Le Fanu’s have the richness and bittersweetness of a man with wider sexual interests. Like Frankenstein or Sherlock Holmes, “Carmilla” may be more famous than its author is. It still appears in horror anthologies, partly because of its theme, partly because it’s probably his best work.
It’s also written more simply than, say, “The Familiar”. You often have to pay attention when you read Le Fanu’s prose:
The mind thus turned in upon itself, and constantly occupied with a haunting anxiety which it dared not reveal, or confide to any human breast, became daily more excited, and, of course, more vividly impressible, by a system of attack which operated through the nervous system; and in this state he was destined to sustain, with increasing frequency, the stealthy visitations of that apparition, which from the first had seemed to possess so unearthly and terrible a hold upon his imagination. (“The Watcher”)
If you don’t concentrate as Le Fanu throws you the words, you drop them and can’t juggle the whirl of metaphor and concept he wants you to experience. The effort required to read his stories is no doubt part of why he isn’t as well-known as he should be. But what you invest is repaid with interest and this collection, in Oxford’s World Classics series, is well represented by the painting on the cover: a detail from the great John Atkinson Grimshaw’s Dulce Domum (1885), with a melancholy-dreaming young woman sitting in a house rich with detail, from peacock feathers to Chinese vases.
More Multi-Magic
The answer, I’m glad to say, is yes. The question is: Can a prime magic-square nest inside a second prime magic-square that nests inside a third prime magic-square? I asked this in Multi-Magic, where I described how a magic square is a square of numbers where all rows, all columns and both diagonals add to the same number, or magic total. This magic square consists entirely of prime numbers, or numbers divisible only by themselves and 1:
43 | 01 | 67 61 | 37 | 13 07 | 73 | 31 Base = 10, magic total = 111
It nests inside this prime magic-square, whose digit-sums in base-97 re-create it:
0619 = [06][37] | 0097 = [01][00] | 1123 = [11][56] 1117 = [11][50] | 0613 = [06][31] | 0109 = [01][12] 0103 = [01][06] | 1129 = [11][62] | 0607 = [06][25] Base = 97, magic total = 1839
And that prime magic-square nests inside this one:
2803 = [1][0618] | 2281 = [1][0096] | 3307 = [1][1122] 3301 = [1][1116] | 2797 = [1][0612] | 2293 = [1][0108] 2287 = [1][0102] | 3313 = [1][1128] | 2791 = [1][0606] Base = 2185, magic total = 8391
I don’t know whether that prime magic-square nests inside a fourth square, but a 3-nest is good for 3×3 magic squares. On the other hand, this famous 3×3 magic square is easy to nest inside an infinite series of other magic squares:
6 | 1 | 8 7 | 5 | 3 2 | 9 | 4 Base = 10, magic total = 15
It’s created by the digit-sums of this square in base-9 (“14 = 15” means that the number 14 is represented as “15” in base-9):
14 = 15 → 6 | 09 = 10 → 1 | 16 = 17 → 8 15 = 16 → 7 | 13 = 14 → 5 | 11 = 12 → 3 10 = 11 → 2 | 17 = 18 → 9 | 12 = 13 → 4 Base = 9, magic total = 39
And that square in base-9 is created by the digit-sums of this square in base-17:
30 = 1[13] → 14 | 25 = 00018 → 09 | 32 = 1[15] → 16 31 = 1[14] → 15 | 29 = 1[12] → 13 | 27 = 1[10] → 11 26 = 00019 → 10 | 33 = 1[16] → 17 | 28 = 1[11] → 12 Base = 17, magic total = 87
And so on:
62 = 1[29] → 30 | 57 = 1[24] → 25 | 64 = 1[31] → 32 63 = 1[30] → 31 | 61 = 1[28] → 29 | 59 = 1[26] → 27 58 = 1[25] → 26 | 65 = 1[32] → 33 | 60 = 1[27] → 28 Base = 33, magic total = 183
126 = 1[61] → 62 | 121 = 1[56] → 57 | 128 = 1[63] → 64 127 = 1[62] → 63 | 125 = 1[60] → 61 | 123 = 1[58] → 59 122 = 1[57] → 58 | 129 = 1[64] → 65 | 124 = 1[59] → 60 Base = 65, magic total = 375
Previously Pre-Posted (please peruse):
The Whisper from the Sea
─But what is that whisper?
─Ah. Then ye hear it?
─Aye. ’Tis thin and eerie, mingling with the waves, and seemeth to come from great distance. I know not the language thereof, but I hear great rage therein.
─As well ye might. We stand near the spot at which the wizard Zigan-Uvalen bested a demon sent against him by an enemy. ’Tis the demon’s whisper ye hear.
─Tell me the tale.
─It is after this wise…
Zigan-Uvalen woke to a stench of brimstone, a crackle of flame, and found himself staring up at a fearsome ebon face, lapped in blood-red fire, horned with curling jet, fanged in razor-sharp obsidian.
“Wake, Wizard!” the apparition boomed. “And make thy peace with thy gods, for I am come to devour thee!”
Zigan-Uvalen sat up and pinched himself thrice.
“Without introduction?” he asked, having verified that he was truly awake.
“Introduction?”
“Well, ’tis customary, in the better magickal circles.”
“Aye? Then know this: I am the Demon Ormaguz, summoned from the hottest corner of the deepest pit of Hell by your most puissant and malicious enemy, the wizard Muran-Egah. I have been dispatched by him over many leagues of plain and ocean to wreak his long-meditated, slow-readied, at-last-matured vengeance on thee.”
“Very well. And what are your qualifications?”
“Qualifications?”
“Aye. Are ye worthy of him who sent you, O Demon Ormaguz?”
“Aye, that I am! And will now dev–”
“Nay, nay!” The wizard raised a supplicatory hand. “Take not offence, O Ormaguz. I ask merely out of form. ’Tis customary, in the better magickal circles.”
“Truly?”
“Truly.”
“Then know this… Well, of formal qualifications, diplomas, and the like, I have none, ’tis true. But I am a demon, thou puny mortal. I have supernatural powers of body and mind, far beyond thy ken.”
“I doubt them not. At least, I doubt not your powers of body, in that ye have travelled so very far and very fast this very night. Or so ye say. But powers of mind? Of what do they consist?”
“Of aught thou carest to name, O Wizard.”
“Then ye have, for instance, much mathematical skill?”
“Far beyond thy ken.”
“How far?”
“Infinitely far, wizard!”
“Infinitely? Then could ye, for instance, choose a number at hazard from the whole and endless series of the integers?”
“Aye, that I could!”
“Entirely at hazard, as though ye rolled a die of infinite sides?”
“Aye! In less than the blink of an eye!”
“Well, so ye say.”
“So I say? Aye, so I say, and say sooth!”
“Take not offence, O Demon, but appearances are against you.”
“Against me?”
“Ye are a demon, after all, unbound by man’s pusillanimous morality.”
“I speak sooth, I tell thee! I could, in an instant, choose a number, entirely at hazard, from the whole and endless series of the integers.”
“And speak it to me?”
“Ha! So that is thy game, wizard! Thou seekest to occupy me with some prodigious number whilst thou makest thy escape.”
“Nay, nay, ye misjudge me, O Demon. Let me suggest this. If ye can, as ye say, choose such a number, then do so and recite its digits to me after the following wise: in the first second, name a single digit – nay, nay, O Demon, hear me out, I pray! Aye, in the first second, name a single digit thereof; in the second second, name four digits, which is to say, two raised to the second power; in the third second, name a number of digits I, as a mere mortal, cannot describe to you, for ’tis equal to three raised to the third power of three.”
“That would be 7,625,597,484,987 digits named in the third second, O Wizard.”
“Ah, most impressive! And your tongue would not falter to enunciate them?”
“Nay, not at all! Did I not tell thee my powers are supernatural?”
“That ye did, O Demon. And in the fourth second, of course, ye would name a number of digits equal to four raised to four to the fourth power of four. And so proceed till the number is exhausted. Does this seem well to you?”
“Aye, very well. Thou wilt have the satisfaction of knowing that ’tis an honest demon who devoureth thee.”
“That I will. Then, O Ormaguz, prove your honesty. Choose your number and recite it to me, after the wise I described to you. Then devour me at your leisure.”
─Then the Demon chose a number at hazard from the whole and endless series of the integers and began to recite it after the wise Zigan-Uvalen had described. That was eighteen centuries ago. The demon reciteth the number yet. That is the whisper ye hear from the sea, which rose long ago above the tomb of Zigan-Uvalen.
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 