Solve $3a^2 + 3ab + b^2 - p^3 = 0$ using infinite descent or otherwise?

Question

I'm trying to get my head around infinite descent proofs for Diophantine equations and I was trying to apply it to a problem, and as you can see, I am struggling with it. Consider the identity $$3a^2 + 3ab + b^2 - p^3 = 0$$

I also know that $a$ and $b$ are coprime, as are $p$ and $b$. I think from this I can infer $a$ and $p$ are coprime too. The trivial solutions that I found are $a = b = p = 0$ and $a = 0, b = p = 1$. But beyond this, I don't think there are further solutions in the positive integers, and I'd like to prove or disprove this.

I think infinite descent might work as a general method here, but I'm struggling with it so far. All I've shown is that $p$ must be odd, and $p = 3K +1$ as neither $p$ nor $b$ is divisible by $3,$ and $p^3\pmod 3\equiv 1$ to match $b^2\pmod 3 \equiv 1$. If anyone knows how to proceed with a proof (or counterproof) of this identity, I would appreciate the pointers!

Answer 1

$$3 \cdot 17^2 + 3 \cdot 17 \cdot 19 + 19^2 - 13^3 = 0$$

Answer 2

taking $$ p = x^2 + 3xy + 3 y^2 $$ with $\gcd(x,y) = 1,$ let us also take $x \neq 0 \pmod 3,$ then

$$ b = x^3 - 9 x y^2 - 9 y^3 \; , \; \; $$ $$ a = 3 x^2 y + 9 x y^2 + 6 y^3 \; . \; \; $$

After which $$b^2 + 3 ba + 3 a^2 = p^3 $$

and we should now fiddle with the gcd's.

The method for finding the above parametrization is Gauss composition, using Dirichlet's method for $x^2 + 3xy + 3 y^2.$ In 3.5 part (b) take $a=a'=1, B=C=3.$ We are cubing the form, so two composition steps are needed. Oh, these are pages 66-67 in first edition of Cox, Primes of the Form $x^2 + n y^2.$ This is given first in Proposition 3.8 on page 49, but there is a typo in the definition of capital $X.$ Also correct in the second edition.

here are some numbers, the first section is with $p$ a prime or prime power (49). The second section need larger $x,y$ to get p divisible by at least two primes.

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x    y    p     b     a         n 
1   14  631 -26459  18270   251239591= 631^3 
1   13  547 -21293  14742   163667323= 547^3 
1   11  397 -13067   9108    62570773= 397^3 
1   10  331  -9899   6930    36264691= 331^3 
1    9  271  -7289   5130    19902511= 271^3 
1    7  169  -3527   2520     4826809= 13^6 
1    6  127  -2267   1638     2048383= 127^3 
1    4   61   -719    540      226981= 61^3 
1    3   37   -323    252       50653= 37^3 
1    2   19   -107     90        6859= 19^3 
1    1    7    -17     18         343= 7^3 
1    0    1      1      0           1=  1  
2   15  769 -34417  24480   454756609= 769^3 
2   11  433 -14149  10296    81182737= 433^3 
2    7  193  -3961   3024     7189057= 193^3 
2    5  109  -1567   1260     1295029= 109^3 
2    3   49   -397    360      117649= 7^6 
2    1   13    -19     36        2197= 13^3 
x    y    p     b     a         n 

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    x    y    p     b     a         n 
    1   15  721 -32399  22320   374805361= 7^3 103^3 
    1   12  469 -16847  11700   103161709= 7^3 67^3 
    1    8  217  -5183   3672    10218313= 7^3 31^3 
    1    5   91  -1349    990      753571= 7^3 13^3 
    2   13  589 -22807  16380   204336469= 19^3 31^3 
    2    9  301  -8011   5940    27270901= 7^3 43^3 
    4   15  871 -38411  29070   660776311= 13^3 67^3 
    4   13  679 -25793  19890   313046839= 7^3 97^3 
    4   11  511 -16271  12870   133432831= 7^3 73^3 
    4    7  247  -4787   4158    15069223= 13^3 19^3 
    5   12  637 -21907  17748   258474853= 7^6 13^3 
    5   11  553 -17299  14256   169112377= 7^3 79^3 
    5    9  403 -10081   8694    65450827= 13^3 31^3 
    5    4  133  -1171   1404     2352637= 7^3 19^3 
    7   10  559 -14957  13770   174676879= 13^3 43^3 
    7    9  481 -11321  10800   111284641= 13^3 37^3 
    8   15 1099 -46063  39330  1327373299= 7^3 157^3 
    8    5  259  -2413   3510    17373979= 7^3 37^3 
    8    1   91    431    270      753571= 7^3 13^3 
   10   11  793 -21869  22176   498677257= 13^3 61^3 
   10    3  217    -53   1872    10218313= 7^3 31^3 
   10    1  133    901    396     2352637= 7^3 19^3 
   11   13 1057 -35173  34632  1180932193= 7^3 151^3 
   11   12  949 -28477  28980   854670349= 13^3 73^3 
   11    6  427  -4177   7038    77854483= 7^3 61^3 
   11    4  301   -829   3420    27270901= 7^3 43^3 
   11    3  247    197   2142    15069223= 13^3 19^3 
   13    9  763 -13841  18414   444194947= 7^3 109^3 
   13    7  589  -6623  11340   204336469= 19^3 31^3 
   13    6  511  -3959   8550   133432831= 7^3 73^3 
   13    2  259   1657   1530    17373979= 7^3 37^3 
   14   15 1501 -55981  57420  3381754501= 19^3 79^3 
   14    9  817 -14023  19872   545338513= 19^3 43^3 
   14    5  481  -1531   6840   111284641= 13^3 37^3 
    x    y    p     b     a         n

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