<s>[INST] How find this diophantine equation integer solution $a^3+b^3=(2ab+1)^2$ 
Find this following diophantine equation  integer solution 
  $$a^3+b^3=(2ab+1)^2$$

I think this equation only have two following solution
$$(a,b)=(1,0),(0,1)$$ 
maybe this equation have no other solution? because can see wolframalpha.com
I think following idea is usefull
$$a^3+b^3=(a+b)(a^2-ab+b^2)=(2ab+1)^2$$
since
$(a+b,a^2-ab+b^2)=1$ so
$a+b,a^2-ab+b^2$ are square numbers?
 [/INST] First look for the solutions of the equation $z^2=a^3+b^3$ which are given for instance in
http://www.cecm.sfu.ca/~nbruin/thesis.pdf, lemma 3.2.6.
Then for each family (give values to $s,t$), in order to find the possibles $a,b$ and check if $2ab+1=z.$ 
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