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Keywords:
Diophantine equation $A^4+nB^4=C^2$; Diophantine equation $A^4-nB^4=C^2$; Diophantine equation $X_1^4+4X_2^4=X_3^8+4X_4^8$; Diophantine equation $Y_1^4=Y_2^4+Y_3^4+4Y_4^4$
Diophantine equation $A^4+nB^4=C^2$; Diophantine equation $A^4-nB^4=C^2$; Diophantine equation $X_1^4+4X_2^4=X_3^8+4X_4^8$; Diophantine equation $Y_1^4=Y_2^4+Y_3^4+4Y_4^4$
Summary:
The two related Diophantine equations: $X_1^4+4X_2^4=X_3^8+4X_4^8$ and $Y_1^4=Y_2^4+Y_3^4+4Y_4^4$, have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.
References:
[1] Choudhry, A.: The Diophantine equation $A^4 + 4B^4 = C^4+4D^4$. Indian J. Pure Appl. Math., 29, 1998, 1127-1128, MR 1672759 | Zbl 0923.11050
[2] Dickson, L. E.: History of the Theory of Numbers. 2, 1952, Chelsea Publishing Company, New York,
[3] Guy, R. K.: Unsolved Problems in Number Theory. 2004, Springer Science+Business Media Inc., New York, Third Edition. MR 2076335 | Zbl 1058.11001
[4] Jena, S. K.: Beyond the Method of Infinite Descent. J. Comb. Inf. Syst. Sci., 35, 2010, 501-511,
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