Diophantine Equation--nth Powers

The 2-1 equation

(1)

is a special case of Fermat's last theorem and so has no solutions for . Lander et al. (1967) give a table showing the smallest for which a solution to

(2)

with is known. An updated table is given below; a more extensive table may be found at Meyrignac's web site.

	1	2	3	4	5	6
2	2
3	3	2
4	3	2
5	4	3
6	7	5	3
7	7	6	5	4
8	8	7	5	5
9	10	9	8	6	5
10	13	12	11	9	7	6

Take the results from the Ramanujan 6-10-8 identity that for , with

(3)

and

(4)

then

(5)

Using

			(6)
			(7)

now gives

(8)

for or 4.