<s>[INST] Prove that for ${a^3} = {b^2} + {c^2}$. where $a$, $b$ and $c$ are positive integers that there are an infinite number of values for $a$. Prove that for ${a^3} = {b^2} + {c^2}$. where $a$, $b$ and $c$ are positive integers that there are an infinite number of values for $a$.
 [/INST] I have got one way of demonstrating it:
Where $z$ is an odd number
Where $n=(z-1)/2$
1). $(10^z)^3=(3•10^{z+n})^2+(10^{z+n})^2$
This means that $10^{3z}= 9•10^{3z-1}+10^{3z-1}$ for $3z-1=2z+2n$ since $n=(z-1)/2$
Therefore $1•10^{3z}=(9/10)10^{3z}+(1/10)10^{3z}$ which is also $10^{3z}(9/10+1/10)$ which is also $1•10^{3z}$
Therefore there are an infinite number of values for z
now if one superimposes what is in the brackets of 1). with $a$, $b$ and $c$ you will get $a^3=b^2+c^2$
Therefore $a$ has an infinity of values
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