How to prove that $a^2+3ab=c^2$ has infinite solutions?

Question

So I got this question from the $2020$ German Math Olympiad (Schul Runde).

The complete translated question is as follows:

In this task, we consider triples $(a, b, c)$ of positive integers and investigate, which of them are solutions of the equation. An example is (1) $2,16,10$ for a,b,c respectively.

$a^2 + 3ab = c^2$

a) Provide two more solutions of (1). b) Show that equation (1) has an infinite number of solutions. c) How many solutions are there if, in addition, $c=2a+3$ applies?

Also, is there a method that I should follow to get numbers for $a,b,c$ and how can i prove it works and that the equation has infinite solutions.

Answer 1

There are also infinitely many primitive solutions (meaning $\gcd(a,b,c)= 1):$ take integers $u,v$ with $\gcd(u,v) = 1 $ and $u \neq 0 \pmod 3.$

Then take (I am insisting $a > 0$ )

Recipe A: $$ a = u^2 \; , \; \; b = 2uv + 3 v^2 \; , \; \; c = u^2 + 3uv $$

Recipe B: $$ a = u^2 +2uv +v^2 \; , \; \; b =u^2 -2uv \; , \; \; c = 2u^2 + uv - v^2 $$

Recipe C: $$ a = 3 u^2 \; , \; \; b = 2uv + v^2 \; , \; \; c = 3u^2 + 3uv $$

The last one needs with $\gcd(u,v) = 1 $ and $v \neq 0 \pmod 3,$ finally $v \neq u \pmod 3,$

Need to check, the middle one needs $ u +v \neq 0\pmod 3.$

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

From Mordell, Diophantine Equations

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

Illustration: I have the computer find solutions by brute force and keep the primitive ones. I have it print the resulting triples from recipes A,B,C, in each case print $u,v$ as the last things on the line.

   a       b       c   recipe      u       v    
   1       0       1     A         1       0
   1       0       1       RAW   
   1     120      19     A         1       6
   1     120      19       RAW   
   1       1       2     B         1       0
   1       1       2       RAW   
   1     133      20     B         7      -6
   1     133      20       RAW   
   1     161      22     A         1       7
   1     161      22       RAW   
   1      16       7     A         1       2
   1      16       7       RAW   
   1     176      23     B         8      -7
   1     176      23       RAW   
   1     208      25     A         1       8
   1     208      25       RAW   
   1      21       8     B         3      -2
   1      21       8       RAW   
   1     225      26     B         9      -8
   1     225      26       RAW   
   1     261      28     A         1       9
   1     261      28       RAW   
   1     280      29     B        10      -9
   1     280      29       RAW   
   1     320      31     A         1      10
   1     320      31       RAW   
   1      33      10     A         1       3
   1      33      10       RAW   
   1     341      32     B        11     -10
   1     341      32       RAW   
   1     385      34     A         1      11
   1     385      34       RAW   
   1      40      11     B         4      -3
   1      40      11       RAW   
   1     408      35     B        12     -11
   1     408      35       RAW   
   1     456      37     A         1      12
   1     456      37       RAW   
   1     481      38     B        13     -12
   1     481      38       RAW   
   1     533      40     A         1      13
   1     533      40       RAW   
   1       5       4     A         1       1
   1       5       4       RAW   
   1     560      41     B        14     -13
   1     560      41       RAW   
   1      56      13     A         1       4
   1      56      13       RAW   
   1     616      43     A         1      14
   1     616      43       RAW   
   1     645      44     B        15     -14
   1     645      44       RAW   
   1      65      14     B         5      -4
   1      65      14       RAW   
   1     705      46     A         1      15
   1     705      46       RAW   
   1     736      47     B        16     -15
   1     736      47       RAW   
   1     800      49     A         1      16
   1     800      49       RAW   
   1     833      50     B        17     -16
   1     833      50       RAW   
   1      85      16     A         1       5
   1      85      16       RAW   
   1       8       5     B         2      -1
   1       8       5       RAW   
   1      96      17     B         6      -5
   1      96      17       RAW   
   3      -1       0     C         1      -1
   3      -1       0       RAW   
   3     143      36     C         1      11
   3     143      36       RAW   
   3     224      45     C         1      14
   3     224      45       RAW   
   3      35      18     C         1       5
   3      35      18       RAW   
   3      80      27     C         1       8
   3      80      27       RAW   
   3       8       9     C         1       2
   3       8       9       RAW   
   4     119      38     B         7      -5
   4     119      38       RAW   
   4      -1       2     B         1       1
   4      -1       2       RAW   
   4      15      14     B         3      -1
   4      15      14       RAW   
   4     175      46     A         2       7
   4     175      46       RAW   
   4     207      50     B         9      -7
   4     207      50       RAW   
   4      39      22     A         2       3
   4      39      22       RAW   
   4      55      26     B         5      -3
   4      55      26       RAW   
   4       7      10     A         2       1
   4       7      10       RAW   
   4      95      34     A         2       5
   4      95      34       RAW   
  12       5      18     C         2       1
  12       5      18       RAW   
  16      11      28     A         4       1
  16      11      28       RAW   
  16       3      20     B         3       1
  16       3      20       RAW   
  16      35      44     B         5      -1
  16      35      44       RAW   
  16      -5       4     A         4      -1
  16      -5       4       RAW   
  25      13      40     A         5       1
  25      13      40       RAW   
  25      -3      20     B         3       2
  25      -3      20       RAW   
  25      -7      10     A         5      -1
  25      -7      10       RAW   
  25       8      35     B         4       1
  25       8      35       RAW   
  25      -8       5     B         2       3
  25      -8       5       RAW   
  27      16      45     C         3       2
  27      16      45       RAW   
  27      -5      18     C         3      -1
  27      -5      18       RAW   
  27       7      36     C         3       1
  27       7      36       RAW   
  27      -8       9     C         3      -2
  27      -8       9       RAW   
  48      -7      36     C         4      -1
  48      -7      36       RAW   
  49     -11      28     A         7      -1
  49     -11      28       RAW   
  49     -15      14     B         3       4
  49     -15      14       RAW   
  49     -16       7     A         7      -2
  49     -16       7       RAW   
  49      -8      35     B         4       3
  49      -8      35       RAW   
   a       b       c   recipe      u       v   

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

Answer 2

Hint: if $a,b,c$ is any solution, show that $na,nb,nc$ is also a solution for any $n\in\mathbb{N}$...

Answer 3

a) $(a,b,c)=(1,1,2), (1,5,4)$. b) $(n,n,2n)$, $n\in\Bbb Z^{+}$ are solutions.c) If $a^2+3ab=c^2$ and $c=2a+3$ then $b=a+4+\frac3a$. So all positive integer solutions in thi case are $(1,8,5),(3,8,9)$.

Answer 4

There are infinitely many primitive solutions with $a=1$:

Take $c \not\equiv 0 \bmod 3$. Then $c^2-1 \equiv 0 \bmod 3$ by Fermat and you get $b$.

Explicitly, $(a,b,c)=(1,3t^2\pm2t,3t\pm1)$ for $t \in \mathbb N^*$ is an infinite family of primitive solutions.