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First let us say what Pell's equation is. We are talking about the indeterminate quadratic equation
Now, although it is fair to say that Brahmagupta was the first to study this equation, it is equally possible to see that earlier authors had studied problems related to Pell's equation. To mention some briefly: Diophantus examines problems related to Pell's equation and we can reduce Archimedes' "cattle problem" to solving Pell's equation although there is no evidence that Archimedes made this connection.
Let us first note that
One property that he deduced was that if (a, b) satisfies Pell's equation so does (2ab, b2 + na2). This follows immediately applying the method of composition to (a, b) and (a, b). Now of course the method of composition can be applied again to (a, b) and (2ab, b2+ na2) to get another solution and Brahmagupta immediately saw that from one solution of Pell's equation he could generate many solutions.
This was not the only way that Brahmagupta used the method of composition. He also noted that, using a similar argument to what we have just given, if x = a, y = b is a solution of nx2 + k = y2 then applying the method of composition to (a, b) and (a, b) gave (2ab, b2 + na2) as a solution of nx2 + k2 = y2 and so, dividing through by k2, gives
How does this help? These values of x and y do not look like integers. Well if k = 2 then, since (a, b) is a solution of nx2 + k = y2 we have na2 = b2 - 2. Thus
y = (b2 + na2)/2 = (2b2 - 2)/2 = b2 - 1
For example, if we attempt to solve 23x2 + 1 = y2 we see that a = 1, b = 5 satisfies 23a2 + 2 = b2 so, by the above argument, x = 5, y = 24 satisfies Pell's equation. Applying the method of composition to the pair (5, 24) gives
Among the examples Brahmagupta gives himself is a solution of Pell's equation
One can only marvel at this brilliant work done by Brahmagupta in 628 AD.
The next step forward was taken by Bhaskara II in 1150. He discovered the cyclic method, called chakravala by the Indians, which was an algorithm to produce a solution to Pell's equation nx2 + 1 = y2 starting off from any "close" pair (a, b) with na2 + k = b2. We can assume that a and b are coprime, for otherwise we could divide each by their gcd and get a "closer" solution with smaller k. Clearly a, k are then also coprime.
The method relies on a simple observation, namely that, for any m, (1, m) satisfies the 'Pell type equation'
Next Bhaskara II knows that there are infinitely many m such that am + b is divisible by k. He chooses the one which makes m2 - n as small as possible in absolute value. If (m2 - n)/k is one of 1, -1, 2, -2, 4, -4 then we can apply Brahmagupta's method to find a solution to Pell's equation nx2 + 1 = y2. If (m2 - n)/k is not one of these values then repeat the process starting this time with the solution x = (am + b)/k, y = (bm + na)/k to the 'Pell type equation' nx2 + (m2 - n)/k = y2 in exactly the same way as we applied the process to na2 + k = b2. Bhaskara II knows (almost certainly by experience rather than by having a proof) that the process will end after a finite number of steps. This happens when an equation of the form nx2 + t = y2 is reached where t is one of 1, -1, 2, -2, 4, -4.
Bhaskara II gives examples in Bijaganita and the first we look at is
Why do we suspect that Bhaskara II had no proof of the method? Well there are at least two reasons. Firstly the proof is long and difficult and would appear to be well beyond 12th Century mathematics. Secondly the algorithm always reaches a solution of Pell's equation after a finite number of steps without stopping when an equation of the type nx2 + k = y2 where k = -1, 2, -2, 4, or -4 is reached and then applying Brahmagupta's method. If experience of the algorithm is only via examples then, knowing how to proceed when k = -1, 2, -2, 4, or -4 is reached, it is natural to switch to Brahmagupta's method at that point. However, when one writes down a proof it should become clear that the algorithm switching to Brahmagupta's method is never necessary (although can reach the solution more quickly).
The next contribution to Pell's equation was made by Narayana who, in the 14th Century, wrote a commentary on Bhaskara II's Bijaganita. Narayana gave some new examples of the cyclic method. Here are two of his examples:
97×72 + 8 = 692
97×202 + 9 = 1972
97×532 + 11 = 5222
97×862 - 3 = 8472
97×5692 - 1 = 56042
We await these solutions, which, if England or Belgic or Celtic Gaul do not produce, then Narbonese Gaul will.Narbonese Gaul, of course, was the area around Toulouse where Fermat lived! One of Fermat's challenge problems was the same example of Pell's equation which had been studied by Bhaskara II 500 years earlier, namely to find solutions to
In Commercium epistolicum Wallis gave two methods of proving Brahmagupta's lemma which are both essentially equivalent to the argument we gave at the beginning of this article based on the result
Wallis published Treatise on Algebra in 1685 and Chapter 98 of that work is devoted to giving methods to solve Pell's equation based on the exchange of letters he had published in Commercium epistolicum in 1658. However, in his algebra text Wallis put all the methods into a standard form.
We should note that by this time several mathematicians had claimed that Pell's equation nx2 + 1 = y2 had solutions for any n. Wallis, describing Brouncker's method, had made that claim, as had Fermat when commenting on the solutions proposed to his challenge. In fact Fermat claimed, correctly of course, that for any n Pell's equation had infinitely many solutions.
Euler gave Brahmagupta's lemma and its proof in a similar form to that which we have given above. He was, of course, aware of the work of Brouncker on Pell's equation as presented by Wallis, but he was totally unaware of the contributions of the Indian mathematicians. He gave the basis for the continued fractions approach to solving Pell's equation which was put into a polished form by Lagrange in 1766. The other major contribution of Euler was in naming the equation "Pell's equation" and it is generally believed that he gave it that name because he confused Brouncker and Pell, thinking that the major contributions which Wallis had reported on as due to Brouncker were in fact the work of Pell.
Lagrange published his Additions to Euler's Elements of algebra in 1771 and this contains his rigorous version of Euler's continued fraction approach to Pell's equation. This established rigorously the fact that for every n Pell's equation had infinitely many solutions. The solution depends on the continued fraction expansion of √n. In the continued fraction of the square root of an integer the same denominators recur periodically. Moreover, the pattern in most of the recurring sequence is "palindromic". i.e. up to the last element, the second half of the periodic sequence is the first half in reverse. The last number in the repeating sequence is double the integer part of the square root.
For example √19 has the continued fraction expansion
which recurs with length 6. The convergent immediately before the point from which it repeats is 170/39 and Lagrange's theory says that
19651490 + 4508361√19
6681448801 + 1532829480√19
2271672940850 + 521157514839√19
772362118440199 + 177192022215780√19
262600848596726810 + 60244766395850361√19
89283516160768675201 + 20483043382566906960√19
References (17 books/articles)
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- H W Lenstra (A pdf file)
Article by: J J O'Connor and E F Robertson