<s>[INST] Extension of Euclid's formula for Pythagorean triples to higher degrees I recently discovered that you can get all the primitive pythagorean triples with this neat formula known as Euclid's formula
$$a = m^2 - n^2, b = 2mn, c = m^2 + n^2$$
Where, m and n are co-prime and each having a different parity, that is one is odd and the other is even.
PrimitivePythagoreanTriples
I am curious to know if it is possible to find such a formula for cubes that is an expression with 2 numbers or multiple numbers $(x_1, x_2, x_3, ...)$ such that setting $ a, b, c, d $ respectively with the expressions satisfies $a^3 + b^3 + c^3 = d^3$
if it does exist can such expressions exist for any integral power n,
such that $ a_1^n + a_2^n + a_3^n + ... + a_n^n = b^n$.
 [/INST] The fourth power case is the Jacobi Madden equation: $a^4+b^4+c^4+d^4=(a+b+c+d)^4$ Jacobi Madden equation.
It gives infinitely many ineteger solutions for $a^4+b^4+c^4+d^4=e^4$ using elliptic curve.
Unfortunately, the parametric solution is not known.
Some small solutions are
$955^4 + 1770^4 + (-2634)^4 + 5400^4  = 5491^4$
$48150^4 + (-31764)^4 + 27385^4 + 7590^4 = 51361^4$
$561760^4 + 1493309^4 + 3597130^4 + (-1953890)^4 = 3698309^4$
For $n=5,$ see Diophantine equation 5th power.
Sastry found $(75v^5-u^5)^5+(u^5+25v^5)^5+(u^5-25v^5)^5+(10u^3v^2)^5+(50uv^4)^5 =(u^5+75v^5)^5$
Other parametric solutions are here.
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