Depends on the Pell equation. For some of them like

you can tell just by looking at them. There is the method of Brahmagupta and Bhaskara two ancient Indian mathematicians.

Generally though, in computational number theory you will require the use of at least a good scientific calculator.

Yes, once you have the fundamental solution ( or any solution other than the trivial one ) you use a recurrence to get more solutions.

You can tell just by looking at them?

By inspection x = 24 and y = 5.

For that one you must compute the answer.

When I say by inspection I mean without computing the convergents. The OP's one requires some computation. The other is as simple  as x +2 = 4.

What did you get?

You used the recurrence to get that. But if you want to get the fundamental solution convergents of the continued fraction is best.

They never come up in an olympiad. The only time someone asks that is when they know nothing and think the problem is easy. Guys like that think everything is as easy as plugging into the quadratic formula.

For instance, find the fundamental solution to:

They could not approach it. It is a problem that is historically famous. This is how the problem begins.

Archimedes wrote:

Page 1

If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who
once upon a time grazed on the ffields of the Thrinacian isle of Sicily, divided into four herds of
different colours, one milk white, another glossy black, the third yellow and the last dappled. In
each herd were bulls, mighty in number according to these proportions: Understand, stranger,
that the white bulls were equal to a half and a third of the black together with the whole of the
yellow, while the black were equal to the fourth part of the dappled and a fifth, together with,
once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were
equal to a sixth part of the white and a seventh, together with all the yellow. These were the
proportions of the cows: The white were precisely equal to the third part and a fourth of the
whole herd of the black; while the black were equal to the fourth part once more of the dappled
and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled
in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the
yellow were in number equal to a sixth part and seventh of the white herd.
If thou canst accurately tell, O stranger, the number of Cattle of the Sun, giving separately
the number of well-fed bulls and again the number of females according to each colour, thou
wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered
among the wise.
But come, understand also all these conditions regarding the cows of the Sun. When the
white bulls mingled their number with the black, they stood rm, equal in depth and breadth,
and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again,
when the yellow and the dappled bulls were gathered into one herd they stood in such a manner
that their number, beginning from one, grew slowly greater till it completed a triangular figure,
there being no bulls of other colours in their midst nor one of them lacking.
If thou art able, O stranger, to find out all these things and gather them together in your
mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast
been adjudged perfect in this species of wisdom.

The equation in post #37 is the Pell equation for that problem.

Read this for more:

http://www.sciencenews.org/sn_arc98/4_1 … thland.htm

It took more that 1800 years for someone to partially solve the problem. It was only solved fully in the 20th century with the aid of computers.

Incidentally, the Pell equation is incorrectly named. Pell had nothing to do with it. It should be called Fermat's equation.

It would seem so. Vardi and others believe he at least formulated the problem correctly even if he was unable to solve for the 205 000 digit answer.

You use two recurrences to find more answers.

Hi;

Yes, that is one way if you have the convergents. x=numerator and y=denominator of the period.

There are other ways that do not require the convergents.

Hi Leroy;

No bother at all. You  remember that the fundamental solution is  x = 10 and y = 3. We use that to get all the rest.

With

Running the recurrences in the forward direction we get the first bunch: