# A Conjecture of Apoloniusz Tyszka on the Addition of Rational Numbers

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This Demonstration illustrates a conjecture of Apoloniusz Tyszka on the set of solutions of a system of linear equations in variables ; the equations are all of the form or for . The conjecture states if such a system has a solution, then there is a solution such that for all .

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Contributed by: Apoloniusz Tyszka (Hugo Kollataj University, Krakow) (March 2011)

Additional work by: Andrzej Kozlowski

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Let be a natural number and denote the set of equations . A conjecture of Apoloniusz Tyszka states that any consistent system has a solution consisting of rationals belonging to . Each consistent system of this kind can be enlarged to a system with a unique solution. This Demonstration considers these enlarged systems.

For , there are only 48 solutions, so they are computed and displayed. If you choose 100 systems, there will therefore be no change when you click the randomize button. For , the total number of solutions is still small enough to be computed (confirming the conjecture in this case) but too large to be displayed. Thus the solutions are computed deterministically but a randomly chosen sample is shown. For and , the number of solutions is too large to compute so a random collection of systems is chosen and their maximal and minimal values are displayed on bar charts.

Note that the system of equations:

always has the solution , containing a number of the maximal conjectured size.

Reference: A. Tyszka, "Some Conjectures on Addition and Multiplication of Complex (Real) Numbers," arXiv:0807.3010.