…ψάμμος ἀριθμὸν περιπέφευγεν… — Πίνδαρου *Ολυμπιόνικος ΙΙ*, 98

“…the sand escapes all numbering…” — Pindar, *Second Olympian Ode*, line 98

…ψάμμος ἀριθμὸν περιπέφευγεν… — Πίνδαρου *Ολυμπιόνικος ΙΙ*, 98

“…the sand escapes all numbering…” — Pindar, *Second Olympian Ode*, line 98

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Papyrocentric Performativity Presents:

• Pocket to Laroussia – *Larousse de Poche* (Librarie Larousse 1954)

• Translated to Heaven – *Les Hommes Volants*, Valerie Moolman, trans. Madeleine Astorkia (Time-Life Books 1981)

• The Eyes of the Infinite Mind – *Ficciones*, Jorge Luis Borges

• Caught by the Furze – *Francis Walker’s Aphids*, John P. Doncaster (British Museum 1961)

• Commit to Crunch – *Maverick Munch: Selecting a Sinisterly Savory Snack to Reinforce Your Rhizomatically Radical Reading*, Will Self (TransVisceral Books 2016)

Or Read a Review at Random: RaRaR

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If primes are like diamonds, powers of 2 are like talc. Primes don’t crumble under division, because they can’t be divided by any number but themselves and one. Powers of 2 crumble more than any other numbers. The contrast is particularly strong when the primes are Mersenne primes, or equal to a power of 2 minus 1:

3 = 4-1 = 2^2 – 1.

4, 2, 1.

7 = 8-1 = 2^3 – 1.

8, 4, 2, 1.

31 = 32-1 = 2^5 – 1.

32, 16, 8, 4, 2, 1.

127 = 2^7 – 1.

128, 64, 32, 16, 8, 4, 2, 1.

8191 = 2^13 – 1.

8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

131071 = 2^17 – 1.

131072, 65536, 32768, 16384, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

524287 = 2^19 – 1.

524288, 262144, 131072, 65536, 32768, 16384, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

2147483647 = 2^31 – 1.

2147483648, 1073741824, 536870912, 268435456, 134217728, 67108864, 33554432, 16777216, 8388608, 4194304, 2097152, 1048576, 524288, 262144, 131072, 65536, 32768, 16384, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

Are Mersenne primes infinite? If they are, then there will be just as many Mersenne primes as powers of 2, even though very few powers of 2 create a Mersenne prime. That’s one of the paradoxes of infinity: an infinite part is equal to an infinite whole.

But are they infinite? No-one knows, though some of the greatest mathematicians in history have tried to find a proof or disproof of the conjecture. A simpler question about powers of 2 is this: Does every integer appear as part of a power of 2? I can’t find one that doesn’t:

0 is in 1024 = 2^10.

1 is in 16 = 2^4.

2 is in 32 = 2^5.

3 is in 32 = 2^5.

4 = 2^2.

5 is in 256 = 2^8.

6 is in 16 = 2^4.

7 is in 32768 = 2^15.

8 = 2^3.

9 is in 4096 = 2^12.

10 is in 1024 = 2^10.

11 is in 1099511627776 = 2^40.

12 is in 128 = 2^7.

13 is in 131072 = 2^17.

14 is in 262144 = 2^18.

15 is in 2097152 = 2^21.

16 = 2^4.

17 is in 134217728 = 2^27.

18 is in 1073741824 = 2^30.

19 is in 8192 = 2^13.

20 is in 2048 = 2^11.

666 is in 182687704666362864775460604089535377456991567872 = 2^157.

1066 is in 43556142965880123323311949751266331066368 = 2^135.

1492 is in 356811923176489970264571492362373784095686656 = 2^148.

2014 is in 3705346855594118253554271520278013051304639509300498049262642688253220148477952 = 2^261.

I’ve tested much higher than that, but testing is no good: where’s a proof? I don’t have one, though I conjecture that all integers do appear as part or whole of a power of 2. Nor do I have a proof for another conjecture: that all integers appear infinitely often as part or whole of powers of 2. Or indeed, of powers of 3, 4, 5 or any other number except powers of 10.

I conjecture that this would apply in all bases too: In any base b all n appear infinitely often as part or whole of powers of any number except those equal to a power of b.

1 is in 11 = 2^2 in base 3.

2 is in 22 = 2^3 in base 3.

10 is in 1012 = 2^5 in base 3.

11 = 2^2 in base 3.

12 is in 121 = 2^4 in base 3.

20 is in 11202 = 2^7 in base 3.

21 is in 121 = 2^4 in base 3.

22 = 2^3 in base 3.

100 is in 100111 = 2^8 in base 3.

101 is in 1012 = 2^5 in base 3.

102 is in 2210212 = 2^11 in base 3.

110 is in 1101221 = 2^10 in base 3.

111 is in 100111 = 2^8 in base 3.

112 is in 11202 = 2^7 in base 3.

120 is in 11202 = 2^7 in base 3.

121 = 2^4 in base 3.

122 is in 1101221 = 2^10 in base 3.

200 is in 200222 = 2^9 in base 3.

201 is in 12121201 = 2^12 in base 3.

202 is in 11202 = 2^7 in base 3.

1 is in 13 = 2^3 in base 5.

2 is in 112 = 2^5 in base 5.

3 is in 13 = 2^3 in base 5.

4 = 2^2 in base 5.

10 is in 1003 = 2^7 in base 5.

11 is in 112 = 2^5 in base 5.

12 is in 112 = 2^5 in base 5.

13 = 2^3 in base 5.

14 is in 31143 = 2^11 in base 5.

20 is in 2011 = 2^8 in base 5.

21 is in 4044121 = 2^16 in base 5.

22 is in 224 = 2^6 in base 5.

23 is in 112341 = 2^12 in base 5.

24 is in 224 = 2^6 in base 5.

30 is in 13044 = 2^10 in base 5.

31 = 2^4 in base 5.

32 is in 230232 = 2^13 in base 5.

33 is in 2022033 = 2^15 in base 5.

34 is in 112341 = 2^12 in base 5.

40 is in 4022 = 2^9 in base 5.

1 is in 12 = 2^3 in base 6.

2 is in 12 = 2^3 in base 6.

3 is in 332 = 2^7 in base 6.

4 = 2^2 in base 6.

5 is in 52 = 2^5 in base 6.

10 is in 1104 = 2^8 in base 6.

11 is in 1104 = 2^8 in base 6.

12 = 2^3 in base 6.

13 is in 13252 = 2^11 in base 6.

14 is in 144 = 2^6 in base 6.

15 is in 101532 = 2^13 in base 6.

20 is in 203504 = 2^14 in base 6.

21 is in 2212 = 2^9 in base 6.

22 is in 2212 = 2^9 in base 6.

23 is in 1223224 = 2^16 in base 6.

24 = 2^4 in base 6.

25 is in 13252 = 2^11 in base 6.

30 is in 30544 = 2^12 in base 6.

31 is in 15123132 = 2^19 in base 6.

32 is in 332 = 2^7 in base 6.

1 is in 11 = 2^3 in base 7.

2 is in 22 = 2^4 in base 7.

3 is in 1331 = 2^9 in base 7.

4 = 2^2 in base 7.

5 is in 514 = 2^8 in base 7.

6 is in 2662 = 2^10 in base 7.

10 is in 1054064 = 2^17 in base 7.

11 = 2^3 in base 7.

12 is in 121 = 2^6 in base 7.

13 is in 1331 = 2^9 in base 7.

14 is in 514 = 2^8 in base 7.

15 is in 35415440431 = 2^30 in base 7.

16 is in 164351 = 2^15 in base 7.

20 is in 362032 = 2^16 in base 7.

21 is in 121 = 2^6 in base 7.

22 = 2^4 in base 7.

23 is in 4312352 = 2^19 in base 7.

24 is in 242 = 2^7 in base 7.

25 is in 11625034 = 2^20 in base 7.

26 is in 2662 = 2^10 in base 7.

1 is in 17 = 2^4 in base 9.

2 is in 152 = 2^7 in base 9.

3 is in 35 = 2^5 in base 9.

4 = 2^2 in base 9.

5 is in 35 = 2^5 in base 9.

6 is in 628 = 2^9 in base 9.

7 is in 17 = 2^4 in base 9.

8 = 2^3 in base 9.

10 is in 108807 = 2^16 in base 9.

11 is in 34511011 = 2^24 in base 9.

12 is in 12212 = 2^13 in base 9.

13 is in 1357 = 2^10 in base 9.

14 is in 314 = 2^8 in base 9.

15 is in 152 = 2^7 in base 9.

16 is in 878162 = 2^19 in base 9.

17 = 2^4 in base 9.

18 is in 218715 = 2^17 in base 9.

20 is in 70122022 = 2^25 in base 9.

21 is in 12212 = 2^13 in base 9.

22 is in 12212 = 2^13 in base 9.

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─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.

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How many blows does it take to demolish a wall with a hammer? It depends on the wall and the hammer, of course. If the wall is reality and the hammer is mathematics, you can do it in three blows, like this:

α’. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.

β’. Γραμμὴ δὲ μῆκος ἀπλατές.

γ’. Γραμμῆς δὲ πέρατα σημεῖα.1. A point is that of which there is no part.

2. A line is a length without breadth.

3. The extremities of a line are points.

That is the astonishing, world-shattering opening in one of the strangest – and sanest – books ever written. It’s twenty-three centuries old, was written by an Alexandrian mathematician called Euclid (fl. 300 B.C.), and has been pored over by everyone from Abraham Lincoln to Bertrand Russell by way of Edna St. Vincent Millay. Its title is highly appropriate: *Στοιχεῖα*, or *Elements*. Physical reality is composed of chemical elements; mathematical reality is composed of logical elements. The second reality is much bigger – infinitely bigger, in fact. In his *Elements*, Euclid slipped the bonds of time, space and matter by demolishing the walls of reality with a mathematical hammer and escaping into a world of pure abstraction.

• Continue reading Neuclid on the Block

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