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 ...
ping
Can this provide a solution for BiXiden Inflation?
I don’t know how many nights I stayed awake trying to figure that out. Good to know it’s finally been solved.
I could have solved those Pell Equations, but I never got a Grant.
No wonder I sucked in algebra!😎
“Integer” sounds racist.
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!
which can be written as x2 – dy2 = 1.
how about x=2, d=3, y=1?
I must be missing something.
I can't tell you how many bar bets I've won because I knew that.
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
Ohhh...yeah that’s mad dumb
..............what?
This article makes my Alabama educated head hurt.
Check out this Advanced Sheep Maths
https://www.youtube.com/watch?v=D2FX9rviEhw&list=PLVo25tMy9RTAPDoqwCH5rrB44btCJAQFj&index=1
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.
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.
Solved it. Write it down and send it in and wait for your Field’s Medal.
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