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
View PDF on arXiv
3 Citations
Highly Influential Citations
1
Background Citations
1

Tables from this paper

3 Citations

Citation Type

Citation Type

The Proof of the Fermat’s Conjecture in the Correct Domain
The distinction between the Domain of Natural Numbers and the Domain of Line gets highlighted. This division provides the new perception to the Fermat’s Conjecture, where to place it and how to prove… Expand
The y-Increment Puzzle, the Law-of-Large-Numbers, and a Second New Proof of Fermat’s Last Conjecture.
This article introduces a second new proof of Fermat’s Last Conjecture (Nwogugu [2020] introduced the first new proof of Fermat’s Last Conjecture) which is studied in the context of Sub-Rings of… Expand
Why did Fermat believe he had ‘a truly marvellous demonstration’ of FLT?
Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary and sufficient for entailing significant mathematical truths only if the proofs are expressed in… Expand

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 also… Expand
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 done… Expand
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 the… Expand
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 method… Expand
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. These… Expand
Number Theory, Volume I: Tools and diophantine equations
  • Graduate Texts in Mathematics,
  • 2007
Une Conjecture de Lebesgue
A Generalization of Fermat’s Last Theorem
...
1
2
...