Diophantine Equation nth Powers

The 2-1 equation

$$A^n+B^n=C^n$$ (1)

is a special case of Fermat's Last Theorem and so has no solutions for $n\geq 3$. Lander et al. (1967) give a table showing the smallest $n$ for which a solution to

$${x_1}^k+{x_2}^k+\ldots+{x_m}^k={y_1}^k+{y_2}^k+\ldots+{y_n}^k,$$

with $1\leq m\leq n$ is known.

	$k$
$m$	2	3	4	5	6	7	8	9	10
1	2	3	3	4	7	8	11	15	23
2	2	2	2	4	7	8	9	12	19
3				3	3	7	8	11	24
4						4	7	10	23
5						5	5	11	16
6								6	27
7									7

Take the results from the Ramanujan 6-10-8 Identity that for $ad=bc$, with

$F_{2m}(a,b,c,d)=(a+b+c)^{2m}+(b+c+d)^{2m}$
$-(c+d+a)^{2m}-(d+a+b)^{2m}+(a-d)^{2m}-(b-c)^{2m}\quad$	(2)

and

$f_{2m}(x,y)=(1+x+y)^{2m}+(x+y+xy)^{2m}$
$-(y+xy+1)^{2m}-(xy+1+x)^{2m}+(1-xy)^{2m}-(x-y)^{2m},\quad$	(3)

then

$$F_{2m}(a,b,c,d)=a^{2m}f_{2m}(x,y).$$ (4)

Using

$$f_2(x,y) = 0$$ (5)

$$f_4(x,y) = 0$$ (6)

now gives

$$(a+b+c)^n+(b+c+d)^n+(a-d)^n = (c+d+a)^n+(d+a+b)^n+(b-c)^n$$ (7)

for $n=2$ or 4.

See also Ramanujan 6-10-8 Identity

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, p. 101, 1994.

Berndt, B. C. and Bhargava, S. ``Ramanujan--For Lowbrows.'' Amer. Math. Monthly 100, 644-656, 1993.

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 653-657, 1966.

Gloden, A. Mehrgradige Gleichungen. Groningen, Netherlands: P. Noordhoff, 1944.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.

Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. ``A Survey of Equal Sums of Like Powers.'' Math. Comput. 21, 446-459, 1967.

Reznick, B. ``Sums of Even Powers of Real Linear Forms.'' Mem. Amer. Math. Soc. No. 463, 96. Providence, RI: Amer. Math. Soc., 1992.