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 ...
The problem looks a bit gay.
All mah integers.
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!
...................
Good little CRT not-sees.
“which can be written as x2 – dy2 = 1.
how about x=2, d=3, y=1?
I must be missing something.”
Exactly ... there has to be some trick to it.
I don’t understand any of it. Don’t have the wattage/brain power for that kind of thought.
What I do notice is that the rate at which mathematicians are finding solutions to ancient mathematical puzzles —has been accelerating in recent decades.
Likely because computers give them power tools for the first time.
You be SMOKIN'!...................
Heh heh... I see what you did there
:)
Glad I’m not the butt of any jokes...
You don’t make an ash of yourself...................
LOL, you’re BRILLIANT!!
I’ll tray.
“which can be written as x2 – dy2 = 1.
how about x=2, d=3, y=1?
I must be missing something.”
Exactly ... there has to be some trick to it.
= = =
Zero is an integer.
For d=0, x=1 for all values of y.
Yeah, I’m not getting something. I think they want solutions for all values of d. Or a general solution of that.
Just fooling around
x2 = 1 + dy2
so, for x=5
25 = 1+ 6 * 2*2
And probably a bunch more simple ones like that.
I guess the problem is trying all the integers from 0 to infinity. That would take a while. I need a Govt. grant for only $1 per integer (from 0 to infinity). That is chicken feed in today’s Govt. budget.
In fact, I’ll only charge $0.00000000000001 per integer from 0 to infinity.
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