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Solve the equation $x^2-5y^2=4$ in positive integers.

I know that the base case is $(3,1)$, but I don't know how to get the general form (like how you do it at a pell equation)

I have the basic solutions $(3,1)$, $(4,2)$, $(11,5)$, $(29,13)$, and $(76,34)$.

The values of $y$ mysteriously correspond to alternating terms in the Fibonacci Sequence [1], 1, [2], 3, [5], 8, [13], 21, [34]. Interesting...

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  • $\begingroup$ Note that you can solve the equation $x^2-5y^2=1 \quad (*)$ and then find solutions for your equation by multiplying the a solution $(x,y)$ of $(*)$ by $2$ for getting a solution of your equation. $\endgroup$
    – flawr
    Commented Jan 2, 2015 at 9:41
  • $\begingroup$ Yes, I know that a solution of $x^2-5y^2=1$ corresponds to a solution of $x^2-5y^2=4$, but not necessarily that the processs is reversible. $\endgroup$
    – user198454
    Commented Jan 2, 2015 at 9:48
  • $\begingroup$ another way is $x/2+\sqrt{5}y/2=(3/2+\sqrt{5}/2)^n$ ($x,y$ are integers, as you'll verify directly (e.g. by setting $n=2$) or by thinking about algebraic integers in $\mathbb Q(\sqrt{5})$) $\endgroup$
    – user8268
    Commented Jan 2, 2015 at 9:49
  • $\begingroup$ @user8268. Thank you! but does this give all the solutions, just like Pell's Equation does? $\endgroup$
    – user198454
    Commented Jan 2, 2015 at 9:50
  • $\begingroup$ Note that you seem to have solutions to $x^2-5y^2=-4$ except for $(3,1)$ which should probably be $(1,1)$ $\endgroup$
    – Mark Bennet
    Commented Jan 2, 2015 at 10:14

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Note that $\frac {1+\sqrt 5}{2}\frac {1-\sqrt 5}2=-1$ so that if $$a^2-5b^2=(a+b\sqrt 5)(a-b\sqrt 5)=\pm4$$ then $\frac {1+\sqrt 5}{2}\frac {1-\sqrt 5}2(a+b\sqrt 5)(a-b\sqrt 5)=\mp 4$ and this means that $$\left(\frac {a+5b}2+\frac{a+b}2\sqrt 5\right)\left(\frac {a+5b}2-\frac{a+b}2\sqrt 5\right)=\mp 4$$

(*)This is obtained by pairing the factors with $+\sqrt 5$ and the factors with $-\sqrt 5$ and gives $$ \left(\frac {a+5b}2\right)^2-5\left(\frac {a+b}2\right)^2=\mp4$$

You then get a sequence $(a,b), (\frac {a+5b}2, \frac {a+b}2), (\frac {3a+5b}2, \frac {a+3b}2) \dots $ where alternating elements give the same sign in the original equation.

The connection with the Fibonacci Sequence is easy to see. You can start with $(a,b)=(1,1)$

The point with this kind of equation is that you can generate additional solutions once you know one solution and some number which is a unit (i.e. a solution to the equation which gives the answer $\pm 1$). What you are really looking for is a fundamental unit - which generates the group of units in the relevant number field. You can then establish a recurrence to find solutions. Here it is $\frac {1+\sqrt 5}2$ which plays that role.

One way of showing that you have all the solutions is to show that you can work back from any solution to a minimum one. If at the stage (*) above you mix the terms so that each term with $+\sqrt 5$ is matched with a term with ${-\sqrt 5}$ then you will find that the solutions decrease.

Looking at the three terms of the sequence noted above $(a_1, b_1)=(a,b); (a_2, b_2)=(\frac {a+5b}2, \frac {a+b}2); (a_3, b_3)=(\frac {3a+5b}2 \frac {a+3b}2)$ it is clear that for any three consecutive terms $$a_1+a_2=a_3$$$$b_1+b_2=b_3$$

This makes it clear that the numbers involved are all integers (once two consecutive integer pairs are found).

The terms in the first place are Lucas Numbers and in the second place are Fibonacci Numbers.

Solutions giving $+4$ are $$(3,1); (7,3); (18,8); (47, 21); (123, 55) \dots $$

Solutions giving $-4$ are $$(1,1); (4,2); (11, 5); (29, 13); (76, 34) \dots$$

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    $\begingroup$ Mark, for your amusement... math.stackexchange.com/questions/742181/… I like the Conway picture method, I put four excerpts with prefix indefinite_binary at zakuski.utsa.edu/~jagy/other.html Dmitry says that the machine Zakuski is being decommissioned in a few weeks, may not have websites after that. Oh, Conway's entire book is available online maths.ed.ac.uk/~aar/papers/conwaysens.pdf $\endgroup$
    – Will Jagy
    Commented Jan 2, 2015 at 19:16
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    $\begingroup$ @WillJagy - I have a copy of the book. Thank you for reminding me of it. I'll have another look - I was taught about quadratic forms by Conway once, long ago, but I think before he developed that particular way of seeing the values. When I read it, I thought it could come in handy. $\endgroup$
    – Mark Bennet
    Commented Jan 2, 2015 at 19:24
  • $\begingroup$ I like it; for small target values, the Lagrange "reduced" forms cycles give all (primitive) representations, Conway is handy for intermediate sized values, as here. I think Stillwell's pages add something valuable as well. $\endgroup$
    – Will Jagy
    Commented Jan 2, 2015 at 19:54
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    $\begingroup$ anyway, if you search the site with string Conway River you get a bunch of my answers with pictures, including recent ones where I show how the form automorphism group generator appears; Conway and Stillwell do not mention that, although it is easy... $\endgroup$
    – Will Jagy
    Commented Jan 2, 2015 at 20:01
  • $\begingroup$ @WillJagy Perhaps there is a subject here for a blog post (if they are still in view)?? $\endgroup$
    – Mark Bennet
    Commented Jan 2, 2015 at 20:05

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