Pell Equation
Pell Equation
Why is the integer equation -n×=1 called the Pell equation?
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In 220 BC, -3×=1 was discovered by Archimedes with methods that have been lost to time.
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1351
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780
In 628 AD, -92×=1 was solved by Brahmagupta, who gave his method.
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1151
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120
In 1150, -61×=1 was solved by Bhāskara II with a general method.
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1766319049
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226153980
In 1657, -313×=1 was given as a challenge problem by Fermat.
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32188120829134849
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1819380158564160
In 1659, Johann Rahn wrote a book that included the method. In 1668, John Pell translated Rahn's book.
Euler thought Pell solved the problem, so he named -n=1 the Pell equation. [1]
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Ignore them. This Demonstration uses the method developed by Lagrange in 1766. His method uses the convergents of continued fractions. For , ,,,,,,,,,,, are the first 12 convergents, each fraction leading to a closer approximation. Of these, ,,,,, give solutions, the last being -3×=1. Lagrange proved that the convergents would always eventually yield solutions. Lagrange also proved that the method by Bhāskara II always works.
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1
1
2
1
5
3
7
4
19
11
26
15
71
41
97
56
265
153
362
209
989
571
1351
780
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1
7
4
26
15
97
56
362
209
1351
780
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1351
2
780