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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>=3. Lander et al. (1967) give a table showing the smallest n for which a solution to

 x_1^k+x_2^k+...+x_m^k=y_1^k+y_2^k+...+y_n^k,
(2)

with 1<=m<=n is known. An updated table is given below; a more extensive table may be found at Meyrignac's web site.

k\m123456
22
332
432
543
6753
77654
88755
9109865
10131211976

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)
(3)

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),
(4)

then

 F_(2m)(a,b,c,d)=a^(2m)f_(2m)(x,y).
(5)

Using

f_2(x,y)=0
(6)
f_4(x,y)=0
(7)

now gives

 (a+b+c)^n+(b+c+d)^n+(a-d)^n 
 =(c+d+a)^n+(d+a+b)^n+(b-c)^n
(8)

for n=2 or 4.


See also

Diophantine Equation, Euler's Sum of Powers Conjecture, Ramanujan 6-10-8 Identity

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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." http://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 x_1^k+...+x_s^k=N (mod p^n)." Math. Comput. 68, 1283-1297, 1999.

Cite this as:

Weisstein, Eric W. "Diophantine Equation--nth Powers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantineEquationnthPowers.html

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