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SOLUTIONS FOR THE FIFTH POWER TAXICAB NUMBER PROBLEM IN $\mathrm{Z}[\sqrt{-2}]$ AND $\mathrm{Z}[\sqrt{-q}]$ WITH $q \equiv 1(\bmod 4)$ A POSITIVE INTEGER PRIME
Keywords:
Taxicab number problem, Diophantine higher degree equationDOI:
https://doi.org/10.17654/0972555525041Abstract
The famous open problem of finding positive integer solutions to $A^5+B^5=C^5+D^5$ is considered, and related solutions are found in the rings $Z[\sqrt{-2}]$ and $Z[\sqrt{-q}]$ with $q \equiv 1(\bmod 4)$ a positive integer prime.
Received: September 1, 2025
Accepted: September 26, 2025
References
[1] G. B. Campbell and A. Zujev, Gaussian integer solutions for the fifth power Taxicab number problem. https://doi.org/10.48550/arXiv.1511.07424.
[2] J. Diaz-Vargas, C. J. Rubio-Barrios and L. G. Santiago-Bonifaz, Eisenstein-Jacobi integer solutions for the fifth power Taxicab number problem, JP Journal of Algebra, Number Theory and Applications 57 (2022), 17-22.
[3] L. E. Dickson, History of the Theory of Numbers, Vol. II, Ch XXII, Originally Published 1919 by Carnegie Inst of Washington, American Mathematical Society 1999, page 644.
[4] G. H. Hardy, Ramanujan, Cambridge University Press, 1927.
[5] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford University Press, London and NY, 1960.
[6] L. J. Mordell, Diophantine Equations, Academic Press, 1969.
[7] I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, India, Wiley, 1991.
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