<s>[INST] How do I see if $a^3+b^3=c^3+d^3$ has any solutions where $ 1 \le a,b,c,d \in \mathbb{Z} \le 1000$ and $a \ne b \ne c \ne d$? How do I see if  $a^3+b^3=c^3+d^3$ has any solutions where $ 1 \le a,b,c,d \in \mathbb{Z} \le 1000$ and $a \ne b \ne c \ne d$ ? 
I know I can write a program to brute force this and find out, but is there a way I can determine this through algebra?
I thought I could use the Difference of Powers formula: 
$(a-c)(a^2+ac+c^2)=a^3-c^3=d^3-b^3=(d-b)(d^2+db+b^2)$. 
At this point I am stuck. 
 [/INST] You are asking about Taxicab Numbers, of which the most famous is
$$1729 = 12^3 + 1^3 = 10^3 + 9^3.$$
There is no (known) algebraic method to solve this equation. All of our current understand of these types of numbers come from computer calculation. There is also an OEIS entry dedicated to them.
If you're curious, we also have found numbers which can be written with $3$+ pairs:
\begin{align*}
    87539319 &=167^3 + 436^3 \\
    &= 228^3 + 423^3 \\
    &= 255^3 + 414^3 \\
    6963472309248 &= 2421^3 + 19083^3 \\
    &= 5436^3 + 18948^3 \\
    &= 10200^3 + 18072^3 \\
    &= 13322^3 + 16630^3. \\
\end{align*}
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