I'm looking for literature on solving problems of the form $$ n_1^\alpha+\cdots+n_k^\alpha=(n_1+\cdots+n_k)^\beta $$ for positive integers $n_1,\ldots,n_k$ and fixed parameters $k$ and $\alpha\ne\beta.$ Any ideas?
This is perhaps similar to multigrade equations or "equal sums of like powers" but not quite similar enough to use those results (as far as I can see). But surely this is too simple not to have been studied?
On
I do not know about the literature, but it is clear that for each $k,$ there is only finitely many solutions. Clearly, $\alpha>\beta.$ Assuming that $n_1\le n_2...\le n_k$ we can estimate, $(kn_k)^{\beta}\ge RHS=LHS\ge n_k^{\alpha},$ hence $n_k^{\alpha-\beta}\le k^{\beta}.$
On
... Continuation:
A list of solutions for certain $(\alpha,\beta)$, $k$:
\begin{array}{|c|l|c|} \hline (\alpha, \beta), \; k & (n_1, \ldots, n_k) & n_1+\ldots+n_k \\ \hline \color{#003388}{(\alpha,\beta)=(4,3), \; k=1} & \color{#CC3300}{1} & \color{#CC3300}{1} \\ \hline \color{#003388}{(\alpha,\beta)=(4,3), \; k=2} & \color{#CC3300}{(4,4)} & \color{#CC3300}{8} \\ \hline \color{#003388}{(\alpha,\beta)=(4,3), \; k=3} & \color{#CC3300}{(9,9,9)} & \color{#CC3300}{27} \\ \hline \color{#003388}{(\alpha,\beta)=(4,3), \; k=4} & (7,14,14,14) & 49 \\ & \color{#CC3300}{(16,16,16,16)} & \color{#CC3300}{64} \\ \hline \color{#003388}{(\alpha,\beta)=(4,3), \; k=5} & (5,15,15,20,20) & 75 \\ & (6,7,12,15,19) & 59 \\ & \color{#CC3300}{(25,25,25,25,25)} & \color{#CC3300}{125} \\ \hline \color{#003388}{(\alpha,\beta)=(4,3), \; k=6} & (2,2,2,8,8,14) & 36 \\ & (2,9,14,21,22,23) & 91 \\ & (3,3,7,13,16,19) & 61 \\ & (4,5,6,11,13,20) & 59 \\ & (28,32,32,32,36,40) & 200 \\ & \color{#CC3300}{(36,36,36,36,36,36)} & \color{#CC3300}{216} \\ \hline \color{#003388}{(\alpha,\beta)=(4,3), \; k=7} & (1,1,2,4,7,14,14) & 43 \\ & (1,2,11,14,17,20,26) & 91 \\ & (1,5,6,6,11,11,21) & 61 \\ & (1,8,12,22,23,28,29) & 123 \\ & (1,22,23,26,29,33,39) & 173 \\ & (2,2,4,12,12,14,22) & 68 \\ & (2,14,14,14,23,23,33) & 123 \\ & (3,11,18,21,26,27,35) & 141 \\ & (3,18,21,27,30,36,36) & 171 \\ & (4,12,14,14,14,28,30) & 116 \\ & (5,7,15,23,23,31,31) & 135 \\ & (5,9,16,20,26,30,33) & 139 \\ & (5,10,11,19,23,24,33) & 125 \\ & (6,7,18,21,25,31,33) & 141 \\ & (6,12,21,32,32,33,35) & 171 \\ & (6,16,16,18,24,34,34) & 148 \\ & (7,7,9,9,12,22,27) & 93 \\ & (7,8,8,8,9,10,25) & 75 \\ & (7,9,9,9,17,23,29) & 103 \\ & (8,11,24,25,32,34,37) & 171 \\ & (9,9,16,17,19,20,35) & 125 \\ & (9,10,22,24,26,32,38) & 161 \\ & (9,12,24,24,30,33,39) & 171 \\ & (10,19,30,33,33,37,43) & 205 \\ & (10,21,21,22,24,31,42) & 171 \\ & (10,21,33,36,39,40,40) & 219 \\ & (13,16,17,26,27,33,41) & 173 \\ & (13,16,19,22,27,37,39) & 173 \\ & (13,17,20,22,24,34,41) & 171 \\ & (13,21,31,35,38,39,44) & 221 \\ & (15,20,24,25,26,32,45) & 187 \\ & (17,17,25,30,33,39,44) & 205 \\ & (17,18,22,31,33,38,44) & 203 \\ & (18,19,24,24,26,38,44) & 193 \\ & (18,26,37,40,41,42,47) & 251 \\ & (19,20,29,31,36,39,47) & 221 \\ & (21,21,25,28,41,41,44) & 221 \\ & (21,28,36,39,39,39,51) & 253 \\ & (22,27,32,36,40,47,47) & 251 \\ & (22,30,30,44,44,44,46) & 260 \\ & (25,25,29,30,35,44,49) & 237 \\ & (25,33,35,40,41,42,53) & 269 \\ & (27,36,37,39,46,46,52) & 283 \\ & \color{#CC3300}{(49,49,49,49,49,49,49)} & \color{#CC3300}{343} \\ \hline \hline \color{#003388}{(\alpha,\beta)=(5,4), \; k=1} & \color{#CC3300}{1} & \color{#CC3300}{1} \\ \hline \color{#003388}{(\alpha,\beta)=(5,4), \; k=2} & \color{#CC3300}{(8,8)} & \color{#CC3300}{8} \\ \hline \color{#003388}{(\alpha,\beta)=(5,4), \; k=3} & \color{#CC3300}{(27,27,27)} & \color{#CC3300}{27} \\ \hline \color{#003388}{(\alpha,\beta)=(5,4), \; k=4} & (22,22,33,44) & 121 \\ & (50,50,60,65) & 225 \\ & \color{#CC3300}{(64,64,64,64)} & \color{#CC3300}{256} \\ \hline \color{#003388}{(\alpha,\beta)=(5,4), \; k=5} & (36,36,36,36,72) & 216 \\ & \color{#CC3300}{(125,125,125,125,125)} & \color{#CC3300}{625} \\ \hline \color{#003388}{(\alpha,\beta)=(5,4), \; k=6} & (1,3,14,21,22,39) & 100 \\ & (3,57,81,96,99,114) & 450 \\ & (4,45,57,90,91,98) & 385 \\ & (5,5,40,45,57,68) & 220 \\ & (7,60,80,85,85,118) & 435 \\ & (10,25,41,55,55,84) & 270 \\ & (14,31,36,44,44,81) & 250 \\ & (16,56,64,64,88,112) & 400 \\ & (18,36,45,72,81,99) & 351 \\ & (21,38,45,75,92,99) & 370 \\ & (25,60,85,115,115,125) & 525 \\ & (30,35,52,77,88,108) & 390 \\ & (30,81,113,115,138,139) & 616 \\ & (34,68,80,110,112,136) & 540 \\ & (35,45,55,75,75,115) & 400 \\ & (36,40,68,82,112,113) & 451 \\ & \color{#CC3300}{(216,216,216,216,216,216)} & \color{#CC3300}{1296} \\ \hline \end{array}
First, note that $\alpha>\beta$ (if we consider $\alpha,\beta \in \mathbb{N}$).
$2$ limitations for $(n_1,\ldots, n_k)$:
A. Using generalized mean (power mean), one can obtain:
$\dfrac{n_1+\ldots+n_k}{k} \leqslant \sqrt[\alpha]{\dfrac{n_1^{\alpha}+\ldots+n_k^{\alpha}}{k}}$,
$\left(\dfrac{n_1+\ldots+n_k}{k} \right)^\alpha \leqslant \dfrac{n_1^{\alpha}+\ldots+n_k^{\alpha}}{k} = \dfrac{(n_1 + \ldots + n_k)^\beta}{k}$,
$\left(n_1+\ldots+n_k \right)^{\alpha-\beta} \leqslant k^{\alpha-1}$.
So, limitation for sum: \begin{array}{|c|} \hline n_1+\ldots+n_k\leqslant k^{\frac{\alpha-1}{\alpha-\beta}}. \\ \hline \end{array}
If we'll consider the case $\beta = \alpha-1$, then $n_1+\ldots+n_k\leqslant k^{\alpha-1}$.
B. We can obtain other (more strong) limitation:
$\dfrac{n_1+\ldots+n_k}{k}\leqslant\sqrt[\beta]{\dfrac{n_1^\beta+\ldots+n_k^\beta}{k}} \leqslant \sqrt[\alpha]{\dfrac{n_1^\alpha+\ldots+n_k^\alpha}{k}}$,
$\dfrac{n_1^\alpha+\ldots+n_k^\alpha}{k^\beta} = \left(\dfrac{n_1+\ldots+n_k}{k}\right)^\beta \leqslant \dfrac{n_1^\beta+\ldots+n_k^\beta}{k} \leqslant \left(\dfrac{n_1^\alpha+\ldots+n_k^\alpha}{k}\right)^{\frac{\beta}{\alpha}}$,
$\left(n_1^\alpha+\ldots+n_k^\alpha\right)^{\frac{\alpha-\beta}{\alpha}} \leqslant k^{\beta - \frac{\beta}{\alpha}} = k^{\frac{(\alpha-1)\beta}{\alpha}}$.
So, limitation for power sum: \begin{array}{|c|} \hline n_1^\alpha+\ldots+n_k^\alpha\leqslant k^{\frac{(\alpha-1)\beta}{\alpha-\beta}}. \\ \hline \end{array}
If we'll consider the case $\beta = \alpha-1$, then $n_1^\alpha+\ldots+n_k^\alpha\leqslant k^{(\alpha-1)\beta}$.
A few examples (for different $n_1, n_2,\ldots,n_k$):
$\color{#003388}{(\alpha,\beta)=(3,2), \;\;k\in \mathbb{N}}$: a famous identity:
$\color{#CC3300}{1^3+2^3+\ldots+k^3 = (1+2+\ldots+k)^2}$.
$\color{#003388}{(\alpha,\beta)=(4,3), \;\;k=5}$:
$6^4+7^4+12^4+15^4+19^4=(6+7+12+15+19)^3$;
$\color{#003388}{(\alpha,\beta)=(4,3), \;\;k=6}$:
$2^4+9^4+14^4+21^4+22^4+23^4=(2+9+14+21+22+23)^3$;
$4^4+5^4+6^4+11^4+13^4+20^4=(4+5+6+11+13+20)^3$;
$\color{#003388}{(\alpha,\beta)=(4,3), \;\;k=7}$:
$(1,2,11,14,17,20,26)$;
$(1,8,12,22,23,28,29)$;
$(1,22,23,26,29,33,39)$;
$(3,11,18,21,26,27,35)$;
$(5,9,16,20,26,30,33)$;
$(5,10,11,19,23,24,33)$;
$(6,7,18,21,25,31,33)$;
$\ldots$
$\color{#003388}{(\alpha,\beta)=(5,4), \;\;k=6}$:
$1^5+3^5+14^5+21^5+22^5+39^5 = (1+3+14+21+22+39)^4$;
$3^5+57^5+81^5+96^5+99^5+114^5 = (3+57+81+96+99+114)^4$;
$4^5+45^5+57^5+90^5+91^5+98^5 = (4+45+57+90+91+98)^4$;
$\ldots$
$\color{#003388}{(\alpha,\beta)=(5,4), \;\;k=7}$:
$1^5+2^5+27^5+38^5+54^5+69^5+80^5 = (1+2+27+38+54+69+80)^4$;
$1^5+5^5+17^5+26^5+34^5+59^5+63^5 = (1+5+17+26+34+59+63)^4$;
$2^5+20^5+23^5+36^5+38^5+43^5+79^5 = (2+20+23+36+38+43+79)^4$;
$3^5+4^5+12^5+16^5+17^5+45^5+48^5 = (3+4+12+16+17+45+48)^4$;
$3^5+18^5+19^5+27^5+35^5+37^5+71^5 = (3+18+19+27+35+37+71)^4$;
$\ldots$