Which pairs of natural numbers $(x, y)$ satisfy $3^x = y^2 + 2$?

Question

As the title says, I have to find all pairs of natural numbers $(x, y)$ such that $3^x = y^2 + 2$ holds. The pairs $(1, 1), (3, 5)$ seem to be the only solutions but I'm not sure how to prove that they indeed are the only pairs which satisfy the given equation.

Answer 1

Elementary proof:

Knowing $x$ needs to be odd, we can write $3^x = 3 w^2$ and ask when $w$ might be a power of three, in $3 w^2 = y^2 + 2.$ This does not happen again after your initial examples, there is an accident: this variable $w$ is frequently divisible by $3,$ but it is not divisible by $9$ unless it is also divisible by $17.$ Look up the phrase Pisano Periods. This is, you see, a finite and definitive check.

Oh, see below, the sequence of $w$ values obeys $$ w_{n+2} = 4 w_{n+1} - w_n $$ It is uncommon for such a law to cover all the $w$ values. Usually it is delayed, as in some $v_{n+8} = 4 v_{n+4} - v_n . $ The necessary thing: $2$ is prime, in our $3 w^2 - 2 $ being a square.

The periodicity in the two lists below can be summarized, and proved as:

$$ w_{n+9} + w_{n} \equiv 0 \pmod 9 $$

$$ w_{n+18} \equiv w_n \pmod 9 $$ and $$ w_{n+9} + w_{n} \equiv 0 \pmod{17} $$

$$ w_{n+18} \equiv w_n \pmod{17} $$

You don't need to find actual values of the sequence $w_n$ to see this: you can just begin with $w_1 = 1, w_2 = 3,$ and reduce either mod 9 or mod 17 each time the rule $ w_{n+2} = 4 w_{n+1} - w_n $ is invoked: mod 9 the sequence is $$ 1, 3, 2, 5, 0, 4, 7, 6, 8, 8, 6, 7, 4, 0, 5, 2, 3, 1, 1, 3, 2, 5, 0, 4, 7, ... $$

mod 17 the sequence is $$ 1, 3, 11, 7, 0, 10, 6, 14, 16, 16, 14, 6, 10, 0, 7, 11, 3, 1, 1, 3, 11, 7, 0, 10, 6, ... $$

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

   1     1  mod nine 1     
   2     3  mod nine 3
   3     11  mod nine 2
   4     41  mod nine 5
   5     153  mod nine 0  ++++++++++   
   6     571  mod nine 4
   7     2131  mod nine 7
   8     7953  mod nine 6
   9     29681  mod nine 8
  10     110771  mod nine 8
  11     413403  mod nine 6
  12     1542841  mod nine 7
  13     5757961  mod nine 4
  14     21489003  mod nine 0  ++++++++++   
  15     80198051  mod nine 5
  16     299303201  mod nine 2
  17     1117014753  mod nine 3
  18     4168755811  mod nine 1
  19     15558008491  mod nine 1
  20     58063278153  mod nine 3
  21     216695104121  mod nine 2
  22     808717138331  mod nine 5
  23     3018173449203  mod nine 0  ++++++++++   
  24     11263976658481  mod nine 4
  25     42037733184721  mod nine 7

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

   1     1  mod seventeen 1
   2     3  mod seventeen 3
   3     11  mod seventeen 11
   4     41  mod seventeen 7
   5     153  mod seventeen 0  ++++++++++   
   6     571  mod seventeen 10
   7     2131  mod seventeen 6
   8     7953  mod seventeen 14
   9     29681  mod seventeen 16
  10     110771  mod seventeen 16
  11     413403  mod seventeen 14
  12     1542841  mod seventeen 6
  13     5757961  mod seventeen 10
  14     21489003  mod seventeen 0  ++++++++++   
  15     80198051  mod seventeen 7
  16     299303201  mod seventeen 11
  17     1117014753  mod seventeen 3
  18     4168755811  mod seventeen 1
  19     15558008491  mod seventeen 1
  20     58063278153  mod seventeen 3
  21     216695104121  mod seventeen 11
  22     808717138331  mod seventeen 7
  23     3018173449203  mod seventeen 0  ++++++++++   
  24     11263976658481  mod seventeen 10
  25     42037733184721  mod seventeen 6

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$