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Apoloniusz Tyszka

  1. A conjecture on integer arithmetic.

    Втр, 11/03/2009 - 16:01 — SomNambul
    Authors: Apoloniusz Tyszka
    Subjects: Logic
    Abstract

    We conjecture: if integers x_1,...,x_n satisfy (x_1)^2>2^(2^n) \vee ...\vee
    (x_n)^2>2^(2^n), then (\forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1))
    \wedge (\forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k))
    \wedge (\forall i,j,k \in {1,...,n} (x_i \cdot x_j=x_k \Rightarrow y_i \cdot
    y_j=y_k)) for some integers y_1,...,y_n satisfying (y_1)^2+...+(y_n)^2>n \cdot
    2^(2^n)+(x_1)^2+...+(x_n)^2.

  2. A hypothetical upper bound for solutions of a Diophantine equation with a finite number of solutions.

    Пнд, 08/17/2009 - 18:30 — SomNambul
    Authors: Apoloniusz Tyszka
    Subjects: Number Theory
    Abstract

    Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}, K \in
    {Z,Q,R,C}. We conjecture that if a system S \subseteq E_n has only finitely
    many solutions in K, then their number does not exceed 2^n. We prove this bound
    for K=C. We construct a system S \subseteq E_{21} such that S has infinitely
    many integer solutions and S has no integer solution in
    [-2^{2^{21-1}},2^{2^{21-1}}]^{21}. We conjecture that if a system S \subseteq
    E_n has a finite number of solutions in K, then each such solution
    (x_1,...,x_n) satisfies (|x_1|,...,|x_n|) \in [0,2^{2^{n-1}}]^n.

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