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设 $r$ 是正整数. 本文运用初等数论方法证明了方程 $((2^{r+1}+1)n)^x+((2^{2r+1}+2^{r+1})n)^y=((2^{2r+1}+2^{r+1}+1)n)^z$适合$(x, y, z)\neq(2, 2, 2)$以及$n>1$的正整数解$(x, y, z, n)$都满足 $x>z>y$; 特别是当 $2^{r}+1$ 是素数时, 该方程仅有正整数解$(x, y, z...
Fermat numbers and integers of the form a k + a l + p
Fermat numbers generalized Fermat numbers Erdos problems Zsigmondy’s theorem covering systems
2015/8/25
In 1849, A. de Polignac [20] conjectured that every odd number larger than 3 can be written as the sum of an odd prime and a power of 2. He found a counterexample 959 soon. In 1934, N. P. Ro- manoff [...
Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers
Mersenne numbers cyclotomic cosets of 2 modulo n Poulet pseudoprime super-Poulet pseudoprime overpseudoprime
2012/6/19
We introduce a new class of pseudoprimes-so called "overpseudoprimes to base $b$", which is a subclass of strong pseudoprimes to base $b$. Denoting via $|b|_n$ the multiplicative order of $b$ modulo $...
Level lowering modulo prime powers and twisted Fermat equations
Modular forms level lowering Diophantine equations
2010/11/26
We discuss a clean level lowering theorem modulo prime powers for weight 2 cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations axn + byn + ...
Applications of variational analysis to a generalized Fermat-Torricelli problem
Variational analysis and optimization generalized Fermat-Torricelli problem
2010/12/1
In this paper we develop new applications of variational analysis and generalized differ-
entiation to the following optimization problem and its specifications: given n closed subsets of a Banach sp...
Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves
Nontrivial algebraic cycles Jacobian varieties of some quotients of Fermat curves
2010/11/26
We obtain the trace map images of the values of certain harmonic vol-umes for some quotients of Fermat curves. These provide the algorithm showing that the algebraic cycles called by k-th Ceresa cycle...
Construction of Algorithms for Optimal Fermat Ray Tracing
Fermat 商中的完全方幂
Fermat商 完全方幂 指数diophantine方程
2012/12/3
设p是奇素数,x和n是大于1的奇数.证明了:当p≡7(mod 12)时,Fermat商F(p,x)不是n次方幂.
Parity considerations in the expansion of Fermat-Pell polynomials
Parity considerations expansion Fermat-Pell polynomials
2010/11/1
For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor})\}$ such that each positive ...