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pro vyhledávání: '"Pink, István"'
Autor:
Pink, István, Sándor, Csaba
Let $n$ be a positive integer. The Diophantine equation $n(x_1+x_2+\dots +x_n)=x_1x_2\dots x_n$, $1 \le x_1\le x_2\le \dots \le x_n$ is called Erd\H{o}s's last equation. We prove that $x_n\to \infty $ as $n\to \infty$ and determine all tuples $(n,x_1
Externí odkaz:
http://arxiv.org/abs/2411.04764
Autor:
Miyazaki, Takafumi, Pink, István
It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. We develop the methods
Externí odkaz:
http://arxiv.org/abs/2403.20037
Autor:
Miyazaki, Takafumi, Pink, István
This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x
Externí odkaz:
http://arxiv.org/abs/2205.11217
Publikováno v:
In Journal of Number Theory September 2024 262:86-102
Autor:
Miyazaki, Takafumi, Pink, István
For any fixed coprime positive integers $a,b$ and $c$ with $\min\{a,b,c\}>1$, we prove that the equation $a^x+b^y=c^z$ has at most two solutions in positive integers $x,y$ and $z$, except for one specific case which exactly gives three solutions. Our
Externí odkaz:
http://arxiv.org/abs/2006.15952
In this work, we give upper bounds for $n$ on the title equation. Our results depend on assertions describing the precise exponents of $2$ and $3$ appearing in the prime factorization of $T_{k}(x)=(x+1)^{k}+(x+2)^{k}+...+(2x)^{k}$. Further, on combin
Externí odkaz:
http://arxiv.org/abs/1709.00400
Akademický článek
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In this paper, we show that there are only finitely many $c$ such that the equation $U_n - V_m = c$ has at least two distinct solutions $(n,m)$, where $\{U_n\}_{n\geq 0}$ and $\{V_m\}_{m\geq 0}$ are given linear recurrence sequences.
Externí odkaz:
http://arxiv.org/abs/1610.06304
Autor:
Pink, István, Ziegler, Volker
Let $u_n$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_1,p_2,\dots,p_s$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $u_n+u_m=w p_1^{z_1} \cd
Externí odkaz:
http://arxiv.org/abs/1604.04720
In this paper, we find all integers $c$ having at least two representations as a difference between a Fibonacci number and a Tribonacci number.
Externí odkaz:
http://arxiv.org/abs/1604.04719