Finding Myself (Okay, Finding i) in p-adic Numbers [UPDATED AND EXPANDED A LOT]
So, I am only just starting to understand p-adic numbers, primarily thanks to 3Blue1Brown and Eric Rowland; it was the latter that pointed out that for some values of p, the square root of negative 1 can be represented.
Mind a splode.
Now, Mr Rowland doesn't say it, but I'm confident in theorizing that the square root of -1 has a representation in p-adic numbers when p is one greater than a multiple of 4. And it should have two, because after all, -1 has two square roots: i and -i.
I find that when we get into wacky stuff -- e.g. analytic continuation, infinite sums, etc -- it helps me to look to a number or series's behavior. Highly unscientific, I'm sure, but it helps.
So how do the square roots of -1 behave? They are numbers that if you multiply them individually by themselves, they equal -1. If you add them together, you get zero. And if you multiply them by each other, you get one.
Naturally I made an Excel spreadsheet. I might have done a large portion of it in a way far more difficult than I needed to, but I didn't have a better idea, so there it was. Anyway, I was able to calculate the p-adic representations of the square roots of -1 for a few values of p, each out to 25 digits. And so:
5-adic:
...1404340423140223032431212
...3040104021304221412013233
13-adic:
...B41505474036C688101550155
...18B7C7858C960644BCB77CB78
17-adic:
...51EG75E81F0160E3D8CGC5A24
...BF209B28F1GFAG2D38404B6ED
(Yes, I italicized the rightmost p digits of the latter two. Yes, I left the 5-adic alone, because there are p^2 digits.)
And yes, we're doing this a la hexadecimal: A is 10, B is 11, and so forth.
Unless I'm mistaken, the next value of p that would allow it would be 29, and doing 25 digits of a 29-adic number just feels weird. Guess I'll have to expand my Excel sheet, huh?
Anyway, that chewed up some of my time this morning, so I thought I'd share it with y'all.
***
You know I did. The 29-adic square roots of negative 1 are:
...M6OH2FGO12DQR8R0E3PO93FGI1C1C
...6M4BQDC4RQF21K1SEP34JPDCARGRH
(There is a zero between the R and the E in the first one; everything else round is an O for 24.)
***
WAIT WAIT HANG ON
So, if p allows the square roots of negative 1 -- aka the fourth roots of unity, along with 1 and -1 -- as long as p can be represented as 4x+1... does that mean that the eighth roots of unity show up when p is 8x+1? That the sixteenth roots show up when p is 16x+1?
Behold: the sixteen 17-adic sixteenth roots of unity (rightmost 17 digits, of course):
Unity:
...00000000000000001
Primitive square root (aka -1):
...GGGGGGGGGGGGGGGGG
Primitive fourth roots (i and -i):
...1F0160E3D8CGC5A24
...F1GFAG2D38404B6ED
Primitive eighth roots (square roots of i and -i):
...G4G6A8A7A51E9C392
...0C0A68696BF274D7F
...1C5C578A12G1F4168
...F4B4B986FE0F1CFA9
Primitive sixteenth roots (fourth roots of i and -i):
...FEA186CC732614E26
...126F8A449DEAFC2EB
...18CG0G5C62EF1G247
...F840G0B4AE21F0ECA
...76CC6GEG1519G5095
...9A44A020FBF70BG7C
...5GB33G39A04B032D3
...B05DD0D76GC5GDE3E
... but would that mean that if p is 3x+1, we could find cube roots of unity?
NO THAT'S ENOUGH MATH FOR TODAY THANK YOU VERY MUCH
***
... okay, so, if they exist in 7-adic numbers, they'd end in 2 and 4. Source: my brain while walking to the toilet just now.
I do, unfortunately, suspect I might end up looking for the 13-adic twelfth roots at some point.
***
OKAY FINE
The 7-adic primitive third roots of unity are:
...31432053116412125443426203642
...35234613550254541223240463024
Now I'm done. For now.
***
So, um.
It is entirely possible to find the 13-adic twelfth roots of unity. My Excel spreadsheet (still giving me 29 digits!) was easily adapted to give me the third roots as well -- see above -- and I was about to go through and get all the third roots of all the fourth roots I worked out before...
Until I realized I only needed one primitive twelfth root, and to multiply it by itself twelve times over.
So:
...115A15C6665B90B4A69B8867923B7
...0B24931093022862A7B4488A5361A
...5A67B41505474036C688101550155
...0B24931093022862A7B4488A53619
...490A9B1B6BB87C521CB9547A8AA6B
...CCCCCCCCCCCCCCCCCCCCCCCCCCCCC
...BB72B70666713C18263144653A916
...C1A839BC39CAA46A25188442796B3
...726518B7C7858C960644BCB77CB78
...C1A839BC39CAA46A25188442796B4
...83C231B16114507AB013785242262
...00000000000000000000000000001
Color coded to display unity, and the primitive second, third, fourth, and sixth roots, respectively.
I'm not gonna say I'm done; I've made that mistake before. I just don't know what I'd do next.
***
small voice: "31 is prime."
me: "Shut up."
***
sigh 31 is prime...
And so, the 31-adic 30th roots of unity (and yes, rightmost 30 digits).
...4AFNHI50C7R1NMGC23OHQET1N2UEK3
...7M8KMB63CR6RCEDNFRCJUNGQJ6BMR9
...JSJ1K5SN1C3E3O2THMK156L6QFSRDR
...DK7D5JJ3LI758DTBCM9G2IOQBMRONJ
...HML4PBGB00LLQBFGFSJAGLI31CM6GQ
...02CC5QEFQD2IP5LM9SAPDFP62F47CG
...602T56H1G0CAKFNLU6J0978CNLA0GH
...9TSSSCB4PMNSPB7NE159JOQLNR7CHK
...9DIIUFRLS54ITDSM6FS4HNQGBE4JIT
...HML4PBGB00LLQBFGFSJAGLI31CM6GP
...QMRJJ89FE56H1E5A7OH7I0R4AC4NND
...T8P8DQAT345E819TEA6B9EMH4ET8K8
...L19R86DCOAKELI4OR9G8EI5ESB9G3O
...QOB5OR8I6HSDKUUAOOIJF51KUM7PQA
...UUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
...QKF7DCPUIN3T78EISR6D4G1T7S0GAS
...N8MA8JORI3O3IGH7F3IB07E4BOJ83M
...B2BTAP27TIRGR6S1D8ATPO9O4F23H4
...HANHPBBR9CNPMH1JI8LESC64J8367C
...D89Q5JEJUU994JFEF2BKE9CRTI8OE5
...USIIP4GF4HSC5P98L2K5HF5OSFQNIF
...OUS1PODTEUIKAF790OBULNMI79KUEE
...L1222IJQ58725JN7GTPLB64973NIDB
...LHCC0F392PQC1H28OF2QD74EJGQBC2
...D89Q5JEJUU994JFEF2BKE9CRTI8OE6
...483BBMLFGPODTGPKN6DNCU3QKIQ77I
...1M5MH4K1RQPGMTL1GKOJLG8DQG1MAN
...9TL3MOHI6KAG9CQ63LEMGCPG2JLER7
...46JP63MCOD2HA00K66CBFPTA08N54L
...000000000000000000000000000001
Once more color-coded, with unity bolded and italicized because I ran out of colors. Otherwise, that's the primitive second, third, fifth, sixth, tenth, and fifteenth roots.
Am I done now?
***
Still a little boggled that, for instance, in 7-adic numbers you can have a representation of - 1/2 +/- root(3)/2 i... but not i itself. It really is a very new way to think about stuff.
***
OH GOD HELP ME I TRIED SOMETHING AND IT WORKED
Here are two 11-adic numbers:
...95900229565801487091712653034
...151AA8815452A9623A19398457A78
These are the two 11-adic numbers that satisfy x^2 = x+1. In other words, THE FUCKING GOLDEN RATIO (and its negative reciprocal).
I cannot. This is too much. I feel as if I've opened Pandora's Box here and just want to seal it shut again.
I could do the silver ratio too
***
I did the silver ratio too
...65536623164112011266421216214
...01130043502554655400245450455
These two 7-adic numbers are the solutions to x^2 = 2x+1.
You know what I'm thinking about now... right?
Why do these values of p allow these... and what values of p would allow both?
***
You can do the golden ratio with 11-adic, 19-adic, and 29-adic.
You can do the silver ratio with 7-adic, 17-adic, and 23-adic.
You can do both with 31-adic.
Well, clearly that's something I need to do.
***
In hopes that my brain will leave this alone for the night, or ideally the weekend, have some 31-adic numbers:
Golden ratio:
…9B66OKD61N20486URIHA481A8U0C6D
…LJOO6AHOT7SUQMO03CDKQMTKM0UIOJ
Silver ratio:
…MBNJE65GUQR1OP6J5S9N78L88ND2R9
…8J7BGOPE043T65OBP2L5NM9MM7HS3O
(Also, I think you can do both golden and silver with 71-adic numbers, but screw that.)
***
lol My brain? Leave me alone? Nah.
So concerning the golden and silver ratios, I made a quick spreadsheet showing me the numbers 1 thru 100 and telling me the square of each number mod p. Why? Because I don't really need to go through and figure out what the ratios are, if I just want to see if it's possible. And that means all I need to do is see if the square root of 5 (golden) or the square root of 2 (silver) can be represented. So, I'm looking for values of p where there's an integer x less than p where x^2 equals 5 or 2 mod p.
Now -- and this is no joke -- while I was typing this, I thought oo what about the bronze ratio and had to update my spreadsheet. The bronze ratio is what satisfies x^2 = 3x+1, of course. So now it's looking for 13 as well.
Anyway, I ran through the primes under 100, and here is what I found:
nothing: 2, 3, 5, 13, 37, 67, 83
golden: 11, 19, 29, 31, 41, 59, 61, 71, 79, 89
silver: 7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97
bronze: 17, 23, 29, 43, 53, 61, 79
golden and silver: 31, 41, 71, 79, 89
golden and bronze: 29, 61, 79
silver and bronze: 17, 23, 79
ALL THREE: 79
And as for numbers that satisfy x^2 = 4x+1, you'll find those everywhere you can do the golden ratio.
***
Fine. Have some 79-adic metallic ratios (using two-digit numbers and hyphens because there aren't enough letters):
golden:
27-60-07-30-41-11-39-21-07-08-60-29-73-45-06-30-57-27-10-60-71-24-66-13-77-08-32-55-44-30
51-18-71-48-37-67-39-57-71-70-18-49-05-33-72-48-21-51-68-18-07-54-12-65-01-70-46-23-34-50
silver:
26-00-10-04-57-23-03-16-25-61-44-13-03-65-55-64-57-60-13-52-70-48-51-47-17-02-33-47-57-10
52-78-68-74-21-55-75-62-53-17-34-65-75-13-23-14-21-18-65-26-08-30-27-31-61-76-45-31-21-71
bronze:
76-73-39-37-50-18-10-14-52-49-68-44-65-40-77-39-35-16-54-74-62-61-10-15-76-77-69-45-05-17
02-05-39-41-28-60-68-64-26-29-10-34-13-38-01-39-43-62-24-04-16-17-68-63-02-01-09-33-73-65
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Catullus 4
Winslow Homer
The Flirt, 1874
Phaselvs ille, quem uidetis, hospites,
This sailing boat, whom you see, guests
This sailin bohut, who yi see, guests,
ait fuisse nauium celerrimus,
it claims that it was once the fastest of the ships,
it clayems that it wuz once the quickest bowut,
neque ullius natantis impetum trabis
nor could any swimming timber
neeva culd eny swimmin timba
nequisse praeterire, siue palmulis
were they able to pass, even with palms
was thih able to pass, even wv parms
opus foret uolare siue linteo.
by flying with that or with canvas.
by flyin wiv tha or with cavnis.
et hoc negat minacis Hadriatici
and he denies that the threatening Adriatic
nd eeh denayes that thi threatenin Adriatic
negare litus insulasue Cycladas
shore can deny, thre Cyc;adic islands
shaw can denaye, te Cycladic ayelands
Rhodumque nobilem horridamque Thraciam
and noble Rhodes and hrace
nd nobull Rhodes and Thrace
Propontida trucemue Ponticum sinum,
with Propontis shivering to the Pontic gulf,
wiv Propontis shiv-a-rin to thi Pontic gulf,
ubi iste post phaselus antea fuit
where he, the boat to be, previously was
weh eeh, the bohut ti bee, previously was
comata silua; nam Cytorio in iugo
hairy wood; for on Cytorus' both
airy wuld; cos on Cytrous bofe
loquente saepe sibilum edidit coma.
often spoke his hair in a loud whistle.
oftin spowk is air inn a lawd wissle.
Amastri Pontica et Cytore buxifer,
Pontic Amastris and Cytorus boxclad,
Pontic Amastris nd Cytorus boxclad,
tibi haec fuisse et esse cognitissima
to you these things were and still are best known
ti yee these things were nd still ar best knawn
ait phaselus: ultima ex origine
the boat claims: from the very first origin
thi bohut claims: from thi very first origin
tuo stetisse dicit in cacumine,
he says that he stood at your summit,
eeh sez that eeh stud at ya sum-it,
tuo imbuisse palmulas in aequore,
your sea in which he dipped his small palms,
ya sea that eeh dipped is smahl palms in tee,
et inde tot per impotentia freta
and from there through so many threatening passages
nd from theh through sow many thretnin ways
erum tulisse, laeua siue dextera
protecting, whether from left or right
protecting, weva from left ah reet
uocaret aura, siue utrumque Iuppiter
the breeze calls, whether juppiter
thi breeze cals, weva jupitter
simul secundus incidisset in pedem;
at once fell on both of his feet;
aal tugeva fell on bowf feet;
neque ulla uota litoralibus deis
nor were there vows to the gods of the shores
neither was vows ti the gods of thi coasts
sibi esse facta, cum ueniret a mari
made, when he came from the sea
maid, wen eeh came from thi sea
nouissimo hunc ad usque limpidum lacum.
towards this very clear lake.
towahds this proper clear layke
sed haec prius fuere: nunc recondita
but these were things of the past: now he grows
but thees were things of thi past: now eeh groes
senet quiete seque dedicat tibi,
in quiet retirement and he devotes himself to you,
in quiet retyament nd eeh devotes imself ti yi,
gemelle Castor et gemelle Castoris.
twin Castor and twin of Castor.
twin Casr-ah and twina Cast-ah.
Translated from Latin (R. A. B. Mynors (ed.), Oxford Classical Texts: C. Valerii Catulli: Carmina) into English and then into Geordie dialect.
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