Generalized Jacobi-Madden equation

Question

The Jacobi-Madden equation $ a^4+b^4+c^4+d^4 = (a+b+c+d)^4 $ has an infinitude of integer solutions with all variables non-zero.

Is it known if the same is true for the general equation $ \sum\limits_{i=1}^n a_i^n = (\sum\limits_{i=1}^n a_i)^n$ ?

I tried to look for some references, but found nothing.

Answer 1

There is a similar question I answered back in 2014. See the MSE post "Integer Solutions of $x^3+y^3+z^3 = (x+y+z)^3.\,$ In summary,

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I. 4th powers

There are infinitely many solutions to,

$$a^4+b^4+c^4+d^4=(a+b+c+d)^4$$

given by the Jacobi-Madden equation (2008).

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II. 5th powers

There are also infinitely many solutions to,

$$a^5+b^5+c^5+d^5+e^5=(a+b+c+d+e)^5$$

given by Lander (1968). See "Geometric Aspects of Diophantine equations involving Equal Sums of Like Powers".

Note: Choudhry, Wroblewski, and others would also give more examples.

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III. 6th powers

In Meyrignac's database http://euler.free.fr/database.txt, there are exactly 177+1 = 178 solutions to,

$$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6+x_7^6 = z^6$$

mostly found between 2009 to 2012. Peter Ansell and others found many of these (6,1,7) and the aim was to find a (6,1,6). Ansell discovered four with the additional property,

$$x_1^6+x_2^6+x_3^6+\dots+x_7^6 = (x_1+x_2+x_3+\dots+x_7)^6$$

But he may have tested only his solutions, so there could be more.

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IV. 7th powers

Choudhry and Wroblewski gave more than 20 solutions to the (7,1,7) that is also,

$$x_1^7+x_2^7+x_3^7+\dots+x_7^7 = (x_1+x_2+x_3+\dots+x_7)^7$$

See this MO link.

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V. 8th powers

There is a (8,1,8) and a (8,1,9), but only the (8,1,10) satisfies,

$$x_1^8+x_2^8+x_3^8+\dots+x_{10}^8\, = \,(x_1+x_2+x_3+\dots+x_{10})^8$$

with $x_i$ as $(6, 16, 34, 58, 66, 142, 152, 171, -184, -226)$.

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VI. 9th powers

There are many solutions to (9,1,9) but Wroblewski found the only one that is also,

$$x_1^9+x_2^9+x_3^9+\dots+x_9^9 = (x_1+x_2+x_3+\dots+x_9)^9$$

with $x_i$ as $(-100, -187, -429, -603, -621, 51, 253, 412, 600)$.

Note: Bremner and Delorme ("On Equal Sums of Ninth Powers") found infinitely many to,

$$x_1^9+x_2^9+x_3^9+\dots+x_{11}^9 = (x_1+x_2+x_3+\dots+x_{11})^9$$

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VII. 10th powers

1. A similar result uses a (10,1,12) by Wroblewski,

$$x_1^{10}+x_2^{10}+x_3^{10}+\dots+x_{12}^{10} = 2^{10}(x_1+x_2+x_3+\dots+x_{12})^{10}$$

The $x_i$ are $(-62, -115, 172, 245, -295, -533, 689, 927, 1011, -1234, -1603, 1684).$

2. Another uses a (10,1,13) by S. Chase,

$$x_1^{10}+x_2^{10}+x_3^{10}+\dots+x_{13}^{10} = (x_1+x_2+x_3+\dots+x_{13})^{10}$$

The $x_i$ are $(-187, -179, -128, -59, -13, -6, 49, 57, 73, 85, 122, 204, 210).$

Answer 2

In Peter Ansell's old website, he has given three four numerical solutions to the equation:

$$a^6+b^6+c^6+d^6+e^6+f^6+g^6=(a+b+c+d+e+f+g)^6=z^6$$

where $(a,b,c,d,e,f,g;z)$ are,

$$(3936,4230,3300,1854,-888,-3187,-4170;\, 5075)\\(3912,3858,2654,3300,-495,-3018,-4884;\, 5327)\\(3546,7209,9194,2046,108,-2712,-8982;\, 10409)\\ (20, 4260, 5778, 11157, 12414, - 7278, - 11994;\, 14357)$$