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
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.
