A New Generalization of Fermat's Last Theorem

@article{Cai2013ANG,
  title={A New Generalization of Fermat's Last Theorem},
  author={Tianxin Cai and Deyi Chen and Yong Zhang},
  journal={arXiv: Number Theory},
  year={2013}
}
In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation \begin{equation} \begin{cases} A+B=C ABC=D^n \end{cases} \end{equation} where $A,B,C,D,n \in\ZZ_{+}$ and $n\geq3$, which may be regarded as a generalization of Fermat's equation $x^n+y^n=z^n$. When $\gcd(A,B,C)=1$, $(1)$ is equivalent to Fermat's equation, which means it has no positive integer solutions. We… Expand

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References

SHOWING 1-10 OF 14 REFERENCES
On the Equations Z
We investigate integer solutions of the superelliptic equation (1) z = F (x, y), where F is a homogenous polynomial with integer coefficients, and of the generalized Fermat equation (2) Ax + By = Cz,Expand
ON SUMS OF PRIMES AND TRIANGULAR NUMBERS
We study whether sufficiently large integers can be written in the form cp + Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number. We alsoExpand
A new variant of the Hilbert-Waring problem
In this paper, we propose a new variant of Waring’s problem: to express a positive integer n as a sum of s positive integers whose product is a k-th power. We define, in a similar way to that doneExpand
Computational Number Theory
Historically, computation has been a driving force in the development of mathematics. To help measure the sizes of their fields, the Egyptians invented geometry. To help predict the positions of theExpand
Ring-Theoretic Properties of Certain Hecke Algebras
The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a methodExpand
Computational Number Theory
TLDR
Computational Number Theory builds the foundation of computational number theory by covering the arithmetic of integers and polynomials at a very basic level and shows how number-theoretic tools are used in cryptography and cryptanalysis. Expand
SOLUTIONS OF THE CUBIC FERMAT EQUATION IN QUADRATIC FIELDS
We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x3 + y3 = z3 in , conditional on the Birch and Swinnerton-Dyer conjecture. TheseExpand
Number Theory, Volume I: Tools and diophantine equations
  • Graduate Texts in Mathematics,
  • 2007
A Generalization of Fermat’s Last Theorem
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