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Experimental Test on the Abc-Conjecture Arno Geimer Under the Supervision of Alexander D

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Experimental Test on the Abc-Conjecture Arno Geimer Under the Supervision of Alexander D Experimental Test on the abc-Conjecture Arno Geimer under the supervision of Alexander D. Rahm, PhD Bachelor thesis at the University of Luxembourg June 2019 Abstract This thesis focuses on finding a counter-example to the abc-Conjecture by com- paring different databases. Contents 1 Introduction 5 1.1 Mathematics and Logic in Ancient Times . .5 1.2 Mathematics and Logic in the Middle Ages . .7 1.3 Mathematics and logic in modern times . .9 1.4 Motivation . 11 2 The abc-Conjecture 12 2.1 The Conjecture . 12 2.2 History of the ABC Conjecture . 13 2.3 Attempted proofs and Consequences . 14 2.3.1 Simpler Proofs of existing Theorems . 15 2.3.2 Conjectures based on the abc-Conjecture . 15 3 The search for a counterexample 16 3.1 Our method . 16 3.1.1 The ABC@Home project and its Databank . 16 3.1.2 The OEIS-Databank . 16 3.1.3 The ABC@Home Algorithm . 17 3.2 Our computations . 18 3.2.1 Getting the files ready . 19 3.2.2 The search algorithm . 20 3.3 Our results . 24 4 Appendix 27 4.1 The conversion algorithm . 27 4.2 Computed results . 30 4.2.1 Sequence 1 . 30 4.2.2 Sequence 2 . 31 4.2.3 Sequence 3 . 32 4.2.4 Sequence 4 . 33 4.2.5 Sequence 5 . 34 4.2.6 Sequence 6 . 35 4.2.7 Sequence 7 . 36 4.2.8 Sequence 8 . 37 4.2.9 Sequence 9 . 38 4.2.10 Sequence 10 . 39 4.2.11 Sequence 11 . 40 4.2.12 Sequence 12 . 41 2 List of Figures 42 3 4 Chapter 1 Introduction The following will be a quick overview of the development of mathematics and logic since its beginning. It will show the history of rigorousness and proofs in mathematics. 1.1 Mathematics and Logic in Ancient Times Several thousand years ago, during the beginning of modern civilisation, math- ematics began as the mere concept of counting, its sole purpose being its usage for trading as well as keeping track of time. One of the earliest artifacts used for mathematical purposes is the Ishango bone which was found in modern- day Congo. It dates back at least 20,000 years and was probably used either as a list of prime numbers or as a six-month lunar calendar.[1] It is believed that the earliest traces of geometry can be found in old Egyptian artifacts from around 5,000 BCE or in monuments in Great Britain from around 3,000 BCE. Nevertheless, these ideas are disputed. It is however undisputed that the old- est known mathematical documents emerged from Babylonian and Egyptian sources. The Babylonians used a numeral system with base 60, complete with multiplication and division, and devel- oped a measure system from as early as 3,000 BCE. It is most astounding that the Babylonians were even able to solve quadratic and cubic equations. The Egyptians probably used a base 10 number system and had mathematical knowledge in both geometry and algebra. For instance, they were able to deter- mine the surface area of three-dimensional objects. However, both the Egyptians and the Babylonians only used mathematics as a tool and thus did not Figure 1.1: Egyptian know the concept of proving their results. The idea numeral system of mathematical concepts having to be proven before being considered to be true only emerged with the independent development of logic in Europe and Asia. In Europe, the study of logic began with Greek philosophers distinguishing between different kinds of sentences, as well as developing techniques for discus- 5 sions based on arguments at around 500 BCE.[2] Even though Pythagoras, as well as Thales, had already established proofs to known statements, the most famous certainly being Pythagoras’ Theorem: In a right triangle, the square of the length of the hypotenuse is the same as the sum of the squares of the two other sides., Greek logic did not yet include the need for proofs. Pythagoras’ proof consists of a simple proof by rearrangement. Around 400 BCE, Plato introduced the notion of definitions and asked about the connection between assumptions and conclusions, which one can view as an early idea of the further notion of a theorem.[3][4] His disciple, Aristotle, was the first to system- atically study logic. In his Organon, he developed logic as a tool for the theoretical sciences, revolv- ing around the notion of deduction. He gives de- tailed ideas about affirmations, denials and contradic- tions and divides proofs into two different categories; those based on "complete" deductions that don’t need proofs, which can be viewed today as axioms, and those based on "incomplete" deductions which need to be proven and follow from the "complete" ones.[5] This is the first description of the concept of a proof. Aristotle also proves that counterexamples disprove a statement. However his school of logic was rivaled by the school of logic, which was developed by the Figure 1.2: Aristotle Stoic philosophers. Both schools differentiate in that Aristotle’s logic is based on "terms"; the subjects and predicates of assertions, while Stoic logic is based on "assertibles"; the equivalent of a proposition.[6] However, not many texts on Stoic logic have sur- vived, as Aristotle’s logic became the dominating form of logic by the end of antiquity. Around 300 BCE, Euclid wrote one of the most important, if not the most important book in the history of mathemat- ics: his Elements. It originally was a collection of 13 books containing definitions, postulates and propo- sitions including their respective proofs in the areas of Euclidean geometry, number theory and algebra. It is universally accepted that Euclid was not the au- thor of most of the theorems and their proofs, but the content of the Elements as well as the mathematical methods used have influenced all of mathematics.[7] Figure 1.3: Euclid His geometric proofs are based on 5 axioms (as cited by wikiversity.org)[8] • A line can be drawn from a point to any other point. • A finite line can be extended indefinitely. • A circle can be drawn, given a center and a radius. • All right angles are ninety degrees. • If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, 6 then the lines meet on that side and not on the other side. These are the five postulates on which Euclidean geometry is based. In Asia, logic developed independently in two different regions: in India and in China. Indian philosophy was divided into several major schools, of which some developed concepts of logic, the most influential being the Nyaya school’s system of logic which later shaped Indian logic.[9] It is based on epistemol- ogy, the philosophy treating the theory of knowledge, and is thus not based on a strict mathematical study of propositions like Western logic. Instead, In- dian logic tries to provide known scientific arguments with rigor. It does not admit statements which are known to be false, which leads to proofs by con- tradiction not being accepted in Indian logic.[10] The most notable Ancient Indian mathematician was Pingala [11], whose work contains the first known description of a binary numeral system as well as a version of the Binomial Theorem which describes the expansion of powers of a binomial.[12] However, since Indian logic was not as well-developed as Greek logic at that moment, In- dian mathematicians did not yet put much rigor into proving their arguments. In China, the development of logic began with the founding of the Mohist school by Mozi around 400 BCE.[4] It was however the intro- duction of Buddhism and the following introduc- tion of Indian logic that the development of Bud- dhist logic appeared in China. As it origi- nated from Indian logic, it is also based on epistemology.[13] Ancient Chinese mathematicians treated the areas of geometry as well as alge- bra, with magic squares dating back as far as 650 BCE and Gaussian elimination being used in The Nine Chapters on the Mathematical Art, an important book in Chinese mathematics fin- Figure 1.4: An excerpt ished around 200 BCE. A detailed commentary from The Nine Chap- by Chinese mathematician Liu Hui from the ters on the Mathemati- 3rd century gave proofs to the problems in the cal Art book.[14] 1.2 Mathematics and Logic in the Middle Ages Around 600 CE, the rise of Islam brought with it Islamic Philosophy and its own take on logic which placed significance on standards of arguments, though this method was quickly overthrown by the Mu’tazili philosophers who favored old Hellenistic philosophy. Several Persian logicians were however dissatisfied with Hellenistic philosophy and corrected it or developed their own forms of logic. Beginning in the 10th century, Islamic scholars began to base their works on clerical authority, which resulted in a decline of Aristotelian logic in Islamic science.[15] The most important aspect of Islamic logic is certainly the development of Avicennian logic which introduced the need for citation as well as inductive logic, i.e. deducting that a conclusion may be true based 7 on some evidence. The period from the 9th to the 15th century is consid- ered to be the Golden Age of Islamic mathematics and saw several Islamic mathematicians proving theorems with the use of arithmetic and algebra.[16] Al-Khwarizmi introduced the Hindu numerical sys- tem, consisting of the numbers 0-10, which proved to be a great tool for improving the efficiency of Is- lamic mathematics.
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