# Mathematics

The project presents the properties of orthodiagonal quadrilaterals, new author problems in respective field of geometry and their solutions. Also, in this project some properties of orthodiagonal quadrilaterals are transferred to octahedron and pyramid. These properties and problems can be used to develop respective products in the field of web design, computer graphics, 3D modeling, engineering and printing on a 3D printer. In addition, the describing properties of orthodiagonal quadrilaterals (including the proposed new solutions of classical problems, also new problems and their solutions), can be applied when studying geometry in schools and in the work of scientific mathematical societies.

## Pythagorean Quintuples

Pythagorean quintuple is a quintuple of five numbers a,b,c,d and e corresponding to the equation a^2 + b^2 + c^2 + d^2 = e^2. In the first part of the research, several different parametrizations of natural Pythagorean quintuples were discovered. Using the parameter selection, a Pythagorean quintuple is obtained, so the generator of Pythagorean pentacles is also called parametrization. The generator that generates all Pythagorean pentacles was discovered and also proved.In the second part of the research, in the set of the whole Pythagorean quintuple the multiplication was defined. It was proved that there is a multiplication unit, and that the multiplication is associative but not commutative. The dismantling of Pythagorean quintuples was also explored.

## Elongated hexapawn

Elongated hexapawn is a board game based on chess, devised by T. R. Dawson. It gained recognision when Martin Gardner used it for the first popular description of machine learning. Still, the game remained unsolved: perfect gameplay was not known. The paper Elongated hexapawn closes this gap, using many combinatorial transformations and the basis of game theory, Sprague-Grundy theorem. Apart from describing the best moves in every position and giving some possibility of using similar methods for solving other board games, there is also a potential application in improving the quality of machine learning. The artificial intelligence can play against the perfect player designed in this paper, effectively measuring the rate of its progress.

## Reduction and visualization of high-dimensional systems

The method described is applicable to quick visualization of high-dimensional systems as well as a general, system-independent model capable of significant optimization in studying large classes of complex systems. With the help of the solution an arbitrary high dimensional system with a number of derivatives studied can be stored like we were only studying 3-dimensional series. Even for small sytems the storage requirement can be decreased with two orders of magnitude. The reconstruction time is less than 0.07 % of that of the time needed for the direct, classical evaluation, this time is quickly decreasing with the number of elements. The model is general enough for answering questions in economy, medicine, ecology, computer science, natural sciences, construction engineering etc.

## Neural Network application to key-point-detection in radiographs

Occasionally, physicians have to measure certain bony structures of a patient precisely and locate so called key points e.g. for purposes such as surgical procedures. Up to now, the radiographs were usually analyzed manually. With increasing frequency, classification tasks are done by a computer in an automated process. In order to automate the key point detection on radiographs, Constantin Tilman Schott developed innovative software that uses artificial intelligence (AI) to identify these important key points. His program uses self-learning algorithms to perform this task. If enough training data is provided, the program can predict the key points with a high degree of accuracy—making the AI as precise as a physician.

## On the Common Prime Divisors of Polynomials

We inspect those polynomials whose coefficients are integers. We say that an integer m is a divisor of a polynomial P if some value of P is divisible by m. Our main result is that the common divisors of any several polynomials are exactly the divisors of a single polynomial. This is extended to prove that the set of prime numbers for which a system of polynomial equations in multiple variables is solvable is exactly the set of prime divisors of some polynomial in one variable. In addition, we prove results on how often a prime number is a common prime divisor of some polynomials – we prove a tight lower bound for this so-called density, and under additional assumptions give a formula for this density. Our work generalizes previous results, and we propose several ideas for further research.

## On the number of points on an algebraic curve in a ring of residues

In our paper we obtained the following results:

1. We introduce a new function R_(f,I) (n), which is the number of all possible combinations of giving coordinates of the solutions of the congruence f(x_1,…,x_k )≡0 mod n, and proved that this function is multiplicative.

2. We obtain the exact number of points on a curve x^m-y^k≡0 modn.

3. We introduce a new definition m/k-power residue modulo n and found an exact formula for their number modulo n. From this formula as a corollary we obtained the full results about m-power residues.

4. We calculated the number of points on a curve ax^2+bxy+cy^2≡0modulo a prime number.

5. We found an exact formula for the number of all possible values of quadratic polynomial mod n.

These results can be useful algebraic geometry and asymmetric cryptography.