Fermat-Like Equations that are not Partition Regular
نویسندگان
چکیده
منابع مشابه
Consistency for partition regular equations
It is easy to deduce from Ramsey’s Theorem that, given positive integers a1, a2, . . . , am and a finite colouring of the set N of positive integers, there exists an injective sequence (xi) ∞ i=1 with all sums of the form ∑m i=1 aixri (r1 < r2 < · · · < rm) lying in the same colour class. The consistency version of this result, namely that, given positive integers a1, a2, . . . , am and b1, b2,...
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A matrix A is said to be partition regular (PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all elements in the same colour class in S, which satisfies Ax = 0. We also say that S is PR for A. Many of the classical theorems of Ramsey Theory, such as van der Waerden’s Theorem and Schur’s Theorem, may naturally be interpreted as statem...
متن کاملSets Partition Regular for n Equations Need Not Solve n + 1
An n ×m rational matrix A is said to be partition regular if for every finite coloring of N there is a monochromatic vector ~x ∈ N with A~x = ~0. A set D ⊆ N is said to be partition regular for A (or for the system of equations A~x = ~0) if for every finite coloring of D there is a monochromatic ~x ∈ D with A~x = ~0. In this paper we show that for every n there is a set that is partition regula...
متن کاملSETS PARTITION REGULAR FOR n EQUATIONS NEED NOT SOLVE n + 1
An n x m rational matrix A is said to be partition regular if for every finite colouring oi \ there is a monochromatic vector x s N with Ax = 0. A set D c f̂J is said to be partition regular for A (or for the system of equations Ax = 0) if for every finite colouring of D there is a monochromatic x e D' with Ax = 6. In this paper we show that for every n there is a set that is partition regular f...
متن کاملSolubility of Fermat Equations
The arithmetic of the equation a1x d 1 + a2x d 2 + a3x d 3 = 0 is considered for d > 2, with the outcome that the set of coefficients for which the equation admits a non-zero integer solution is shown to have density zero.
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ژورنال
عنوان ژورنال: Combinatorica
سال: 2017
ISSN: 0209-9683,1439-6912
DOI: 10.1007/s00493-016-3640-2