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.

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: Dover, pp. 653-657, 2005.

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.

Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." https://euler.free.fr.

Reznick, B. Sums of Even Powers of Real Linear Forms. Providence, RI: Amer. Math. Soc., 1992.

Sekigawa, H. and Koyama, K. "Nonexistence Conditions of a Solution for the Congruence ." Math. Comput. 68, 1283-1297, 1999.