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Given two sequences

$$a(n)=4a(n-1)-a(n-2), \quad a(1)=1,\; a(2)=5;$$ $$b(n)=10b(n-1)-b(n-2), \quad b(1)=1,\; b(2)=11.$$

If some $a(n)=b(m)$, $m,n\in\mathbb{N}$, exist?(Except a(1)=b(1) ) If there are no common items, how to prove it?

The Two solution equation follows from wolfram alpha.

Thanks a lot, I love you master!

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    $\begingroup$ Dear Antin Grudkin ,Thanks for ur editing and correction. $\endgroup$ – Alex Jun 14 '18 at 14:45
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Linear difference equations are much like differential equations

To solve

$$ a_n = 4 a_{n-1}-a_{n-2},\: a_1=1,\; a_2 = 5 $$

we assume that $a_n = \alpha^n$ and substituting results in

$$ (1-4\alpha+\alpha^2)\alpha^n = 0 $$

now solving

$$ 1-4\alpha+\alpha^2 = 0 $$

gives

$$ a_n = c_1(-2-\sqrt5)^n+c_1(-2+\sqrt5)^n $$

Analogously for

$$ b_n = c_3(5-2\sqrt6)^2+c_4(5+\sqrt6)^n $$

Considering the initial conditions we have

$$ a_n =-\frac{\left(1+\sqrt{3}\right) \left(2 \left(2-\sqrt{3}\right)^n+\sqrt{3} \left(2-\sqrt{3}\right)^n-\left(2+\sqrt{3}\right)^n\right)}{2 \left(2+\sqrt{3}\right)} = \frac{1}{2} \left(1+\sqrt{3}\right) \left(\left(2+\sqrt{3}\right)^{n-1}-\left(2-\sqrt{3}\right)^n\right) $$

and

$$ b_n = -\frac{\left(2+\sqrt{6}\right) \left(5 \left(5-2 \sqrt{6}\right)^n+2 \sqrt{6} \left(5-2 \sqrt{6}\right)^n-\left(5+2 \sqrt{6}\right)^n\right)}{4 \left(5+2 \sqrt{6}\right)} = \frac{1}{4} \left(2+\sqrt{6}\right) \left(\left(5+2 \sqrt{6}\right)^{n-1}-\left(5-2 \sqrt{6}\right)^n\right) $$

I hope this helps.

NOTE

$$ a_n = \eta_n + \gamma_n\sqrt 3\\ b_n =\xi_n+\mu_n\sqrt6 $$

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  • $\begingroup$ Thaks...Its a misunderstanding. I already know how to get the solution.Its not my main question. I just need to find the common items in the two sequences.....see? $\endgroup$ – Alex Jun 14 '18 at 16:07
  • $\begingroup$ To handle that question it is a note at the bottom. The way to get there is to show, perhaps inductively that $\gamma_n \ne 0$ and $\mu_n\ne 0$ $\endgroup$ – Cesareo Jun 14 '18 at 16:11
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    $\begingroup$ If $n\in\mathbb{N}$, $$ \gamma_n=0 , \mu_n=0 $$ $\endgroup$ – Alex Jun 14 '18 at 16:34
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I do not think this is elementary. For your sequence called $a_n \; ,$ you are giving the $x$ values in the Pell type $$ x^2 - 3 y^2 = -2. $$ For your sequence called $b_n \; ,$ you are giving the $x$ values in the Pell type $$ 2x^2 - 3 z^2 = -1. $$ Multiply and subtract, whenever there is a common $x$ value, $$ z^2 - 2 y^2 = -1. $$

There are results, for example in Mordell's book, on pairs of Pell type equations. I will see if I can find anything. This question has some references (but no final resolution) A System of Simultaneous Pell Equations

Alright, SZALAY gives an algorithm for completely solving this sort of thing.

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  • $\begingroup$ Yes ,its pell, so its integer solution x, y are second order linear recursive..the two equation you give , is my original question . I just ask by recursive type, thats all, thank you so much. $\endgroup$ – Alex Jun 14 '18 at 17:04
  • $\begingroup$ @Alex Again, I do not think this can be finished by elementary means. Szalay (Appendix, page 84) lists five similar problems solved before 1950 where we can assume no computers were used. Perhaps your version is also solved in one of the indicated articles. For you, though, there is no way without computer help on, for example, elliptic curves or quartic Thue equations. $\endgroup$ – Will Jagy Jun 14 '18 at 18:27
  • $\begingroup$ Will Jagy, I used GSP' GGB' Mathmatica' even C language with mod function or graphic , try to find some rule, but I failed. Anyway, thanks for ur advice.. $\endgroup$ – Alex Jun 14 '18 at 19:15
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I did not realize that this problem moves naturally into my area, ternary quadratic forms. The outcome is that the overlap you have already found is the only one. I found out this simple technique from reading Szalay. Oh, it all reduces to confirming that there are only trivial integer solutions to a quartic, I put the information and a request for confirmation at Quartic: using a Thue tool truly

Alright, we have $$ x^2 - 3 y^2 = -2$$ and $$ 2x^2 - 3 z^2 = -1$$ Multiply these by $-1$ and $2$ to get $$ -x^2 + 3 y^2 = 2 \; , $$ $$ 4 x^2 - 6 z^2 = -2 \; , $$ so $$ 3x^2 + 3 y^2 - 6 z^2 = 0 \; , $$ finally $$ x^2 + y^2 = 2 z^2. $$ Since we can see from the original Pell type equations that $z$ must be odd, meanwhile $\gcd(x,y,z) = 1.$

Here is the part related to Pythagorean triples, but adjusted for $x^2 + y^2 = 2 z^2$: all (primitive) triples can be formed by $$ x = r^2 + 2rs - s^2 \; , $$ $$ y = r^2 - 2rs - s^2 \; , $$ $$ z = r^2 + s^2. $$ Actually, for any of the three variables you can choose $\pm \; .$ See Parametric characterization for $x^2 + y^2 = 2z^2$

Plugging back in, either to $x^2 - 3 y^2 = -2$ or $2 x^2 - 3 z^2 = -1,$ we get the (integer) equation $$ r^4 - 8 r^3 s + 2 r^2 s^2 + 8 r s^3 + s^4 = 1. $$ This is called a Thue equation. I ran it in gp-Pari, the only solutions are $$ (1,0) \; ; \; (-1,0) \; ; \; (0,1) \; ; \; (0,-1) $$ so that the $x,y,z$ triples can only be made up of $\pm 1 \; .$

I also ran it in the online MAGMA calculator. The syntax in June 2018 has updated compared with published articles. enter image description here

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  • $\begingroup$ Pell --> Pythagorean triples -->Thue ,maybe I hv to realize "what is Thue equation" , OHCH Thanks $\endgroup$ – Alex Jun 15 '18 at 5:01

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