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Mathematicians Crack a Simple but Stubborn Class of Equations [Pell Equations]
Quanta Magazine ^ | August 10, 2022 | Jordana Cepelewicz

Posted on 08/10/2022 3:30:05 PM PDT by BenLurkin

Archimedes posed a riddle about herding cattle... His problem ultimately boiled down to an equation that involves the difference between two squared terms, which can be written as x2 – dy2 = 1. Here, d is an integer — a positive or negative counting number — and Archimedes was looking for solutions where both x and y are integers as well.

This class of equations, called the Pell equations, has fascinated mathematicians over the millennia since.

Indian mathematician Brahmagupta, and later the mathematician Bhāskara II, provided algorithms to find integer solutions to these equations. In the mid-1600s, the French mathematician Pierre de Fermat ...discovered that in some cases, even when d was assigned a relatively small value, the smallest possible integer solutions for x and y could be massive. When he sent a series of challenge problems to rival mathematicians, they included the equation x2 – 61y2 = 1, whose smallest solutions have nine or 10 digits. (As for Archimedes, his riddle essentially asked for integer solutions to the equation x2 – 4,729,494y2 = 1. “To print out the smallest solution, it takes 50 pages,”

But the solutions to the Pell equations can do much more. For instance, say you want to approximate 2–√, an irrational number, as a ratio of integers. It turns out that solving the Pell equation x2 – 2y2 = 1 can help you do that: 2–√ (or, more generally, d−−√) can be approximated well by rewriting the solution as a fraction of the form x/y.

[T]hose solutions also tell you something about particular number systems, which mathematicians call rings. In such a number system, mathematicians might adjoin 2–√ to the integers. Rings have certain properties, and mathematicians want to understand those properties. The Pell equation, it turns out, can help them do so.

(Excerpt) Read more at quantamagazine.org ...

TOPICS: Miscellaneous
KEYWORDS: archimedes; epigraphyandlanguage; godsgravesglyphs; greece; pellequations
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1 posted on 08/10/2022 3:30:05 PM PDT by BenLurkin

To: SunkenCiv

ping

2 posted on 08/10/2022 3:30:23 PM PDT by BenLurkin (The above is not a statement of fact. It is either opinion, or satire, or both.)

To: BenLurkin

Can this provide a solution for BiXiden Inflation?

3 posted on 08/10/2022 3:32:43 PM PDT by Paladin2

To: BenLurkin

I don’t know how many nights I stayed awake trying to figure that out. Good to know it’s finally been solved.

4 posted on 08/10/2022 3:35:29 PM PDT by Hot Tabasco (Don't walk thru the watermelon patch)

To: Hot Tabasco
Interesting, but they just had to use the BS "counting number" as well as the perfectly acceptable "integer".

It's a wonder kids today can do any math at all.
5 posted on 08/10/2022 3:38:13 PM PDT by BikerJoe

To: BenLurkin

I could have solved those Pell Equations, but I never got a Grant.

6 posted on 08/10/2022 3:45:10 PM PDT by ClearCase_guy (We are already in a revolutionary period, and the Rule of Law means nothing. It's "whatever".)

To: BenLurkin

No wonder I sucked in algebra!😎

7 posted on 08/10/2022 3:45:18 PM PDT by Bonemaker (invictus maneo)

To: BikerJoe

“Integer” sounds racist.

8 posted on 08/10/2022 3:56:17 PM PDT by ViLaLuz (2 Chronicles 7:14)

To: ViLaLuz
“Integer” sounds racist.

OK, now I'm triggered :-)
9 posted on 08/10/2022 4:02:01 PM PDT by BikerJoe

To: ViLaLuz

I hope that the math team included a black, a hispanic, an indigenous person, a gay, a lesbian and a trans person.

Otherwise the results cannot be considered “knowledge” but must instead be treated as heretical thought and banned!

10 posted on 08/10/2022 4:06:37 PM PDT by cgbg (A kleptocracy--if they can keep it. Think of it as the Cantillon Effect in action.)

To: BenLurkin

which can be written as x2 – dy2 = 1.

I must be missing something.

11 posted on 08/10/2022 4:10:38 PM PDT by Scrambler Bob (My /s is more true than your /science (or you might mean /seance))

To: BenLurkin
For instance, say you want to approximate 2–√, an irrational number, as a ratio of integers. It turns out that solving the Pell equation x2 – 2y2 = 1 can help you do that: 2–√ (or, more generally, d−−√) can be approximated well by rewriting the solution as a fraction of the form x/y.

I can't tell you how many bar bets I've won because I knew that.

12 posted on 08/10/2022 4:23:38 PM PDT by GreenHornet

To: BenLurkin

wow and before calculators and computers
As obama said -— you think youre smart????, theres a lot of smart people-—
Lavish with —
Derisive inflection, smirk and clown smile -—
Hats off to mathimagicians

13 posted on 08/10/2022 4:39:35 PM PDT by Recompennation (Don’t blame me my vote didn’t count so mee )

To: BikerJoe

14 posted on 08/10/2022 4:43:37 PM PDT by one guy in new jersey

To: BenLurkin

15 posted on 08/10/2022 4:48:45 PM PDT by SamAdams76 (3,703,267 users on Truth Social)

To: BenLurkin

..............what?

16 posted on 08/10/2022 4:49:41 PM PDT by military cop (I carry a .45....cause they don't make a .46....)

Check out this Advanced Sheep Maths

17 posted on 08/10/2022 4:53:10 PM PDT by algore

To: BenLurkin; StayAt HomeMother; Ernest_at_the_Beach; 1ofmanyfree; 21twelve; 24Karet; ...
Thanks BenLurkin. If they can be solved, and the Mell equations can be solved, a lot of things will fall into place pell mell. Yeah, my heart wasn't in that one.

18 posted on 08/10/2022 4:55:21 PM PDT by SunkenCiv (Imagine an imaginary menagerie manager imagining managing an imaginary menagerie.)

Can this provide a solution for BiXiden Inflation?

Yes. The solution is a total tax intake that can be shown to be no less than 25 digits. Inflation is solved by blowing up every economy in the universe.

19 posted on 08/10/2022 5:38:10 PM PDT by AndyJackson

To: Scrambler Bob

Solved it. Write it down and send it in and wait for your Field’s Medal.

20 posted on 08/10/2022 5:40:33 PM PDT by AndyJackson