• Journal of the Korea Institute of Information Security & Cryptology
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• v.20 no.3
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• pp.3-8
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• 2010

There are some efficient brute force algorithm for factorization. Fermat's factorization is one of the way of brute force attack. Fermat's method works best when there is factor near the square-root. This paper shows that why Fermat's method is effective and verify that there are only one answer. Because there are only one answer, we can start Fermat's factorization anywhere. Also, we convert from factorization to finding square number.

• Korean Journal of Mathematics
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• v.18 no.1
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• pp.29-35
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• 2010

A Fermat-type equation deals with representing a nonzero constant as a sum of kth powers of nonconstant functions. Suppose that $k{\geq}2$. Consider $\sum_{i=1}^{p}\;f_i(z)^k=1$. Let p be the smallest number of functions that give the above identity. We consider the Fermat-type equation for MAobius transformations and obtain $k{\leq}p{\leq}k+1$.

• The Pure and Applied Mathematics
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• v.26 no.2
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• pp.85-97
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• 2019

In this paper we introduce the reciprocal-negative Fermat's equation induced by the famous equation in the Fermat's Last Theorem, establish the general solution in the simplest cases and the differential solution to the equation, and investigate, then, the generalized Hyers-Ulam stability in a $quasi-{\beta}-normed$ space with both the direct estimation method and the fixed point approach.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.609-616
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• 2021

We study the solvability of the Fermat's matrix equation in some classes of 2-by-2 matrices. We prove the Fermat's matrix equation has infinitely many solutions in a set of 2-by-2 positive semidefinite integral matrices, and has no nontrivial solutions in some classes including 2-by-2 symmetric rational matrices and stochastic quadratic field matrices.

• Korean Journal of Mathematics
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• v.27 no.1
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• pp.81-92
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• 2019

We introduce a reciprocal-negative Fermat's equation generalized with constants coefficients and investigate its stability in a quasi-${\beta}$-normed space.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1845-1858
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• 2018

This paper is to consider the generalized Fermat difference equations with different types which ever considered by Li [14], Ishizaki and Korhonen [9], Zhang [26] and Liu [15-18], respectively. Some new observations and results on these equations will be given.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.983-1002
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• 2021

We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations [f(z)f'(z)]ⁿ + P²(z)f^m(z + 𝜂) = Q(z) and [f(z)f'(z)]ⁿ + P(z)[∆_𝜂f(z)]^m = Q(z), where P(z) and Q(z) are non-zero polynomials, m and n are positive integers, and 𝜂 ∈ ℂ \ {0}. In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation P²(z) [f^(k)(z)]² + [αf(z + 𝜂) - 𝛽f(z)]² = e^r(z), where $P(z){\not\equiv}0$ is a polynomial, r(z) is a non-constant polynomial, α ≠ 0 and 𝛽 are constants, k is a positive integer, and 𝜂 ∈ ℂ \ {0}. Our results generalize some previous results.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.405-417
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• 2021

The aim of this paper is to provide the residues of Euler polynomials modulo p² in terms of alternating sums of like powers of numbers in arithmetical progression. Also, we establish the analogue of a classical congruence of Lehmer.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.83-99
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• 2022

In this paper, we extend some previous works by Liu et al. on the existence of transcendental entire solutions of differential-difference equations of Fermat type. In addition, we also present a precise description of the associated entire solutions.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.447-456
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• 2015

In this paper, we investigate the differential-difference equation $(f(z+c)-f(z))^2+P(z)^2(f^{(k)}(z))^2=Q(z)$, where P(z), Q(z) are nonzero polynomials. In addition, we also investigate Fermat type q-difference differential equations $f(qz)^2+(f^{(k)}(z))^2=1$ and $(f(qz)-f(z))^2+(f^{(k)}(z))^2=1$. If the above equations admit a transcendental entire solution of finite order, then we can obtain the precise expression of the solution.