Abstract
Number theory is the branch of mathematics that is concerned with the mathematical properties of the natural numbers and integers. These include properties such as the parity of a number; divisibility; additive and multiplicative properties; whether a number is prime or composite; the prime factors of a number; the greatest common divisor and least common multiple of two numbers; and so on. Number theory has many applications in computing: for example, the security of the RSA public key cryptographic system relies on the infeasibility of the integer factorization problem for large numbers. There are several unsolved problems in number theory: e.g., Goldbach’s Conjecture states that every even integer greater than two is the sum of two primes.
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Notes
- 1.
Goldbach was an eighteenth century German mathematician and Goldbach’s conjecture has been verified to be true for all integers n < 12 × 1017.
- 2.
Pierre de Fermat was a 17th French civil servant and amateur mathematician. He occasionally wrote to contemporary mathematicians announcing his latest theorem without providing the accompanying proof and inviting them to find the proof. The fact that he never revealed his proofs caused a lot of frustration among his contemporaries, and in his announcement of his famous last theorem he stated that he had a wonderful proof that was too large to include in the margin. He corresponded with Pascal and they did some early work on the mathematical rules of games of chance and early probability theory. He also did some early work on the Calculus.
- 3.
Pythagoras of Samos (a Greek island in the Aegean sea) was an influential ancient mathematician and philosopher of the sixth century B.C. He gained his mathematical knowledge from his travels throughout the ancient world (especially in Egypt and Babylon). He became convinced that everything is number and he and his followers discovered the relationship between mathematics and the physical world as well as relationships between numbers and music. On his return to Samos he founded a school and he later moved to Croton in southern Italy to set up a school. This school and the Pythagorean brotherhood became a secret society with religious beliefs such as reincarnation and they were focused on the study of mathematics. They maintained secrecy of the mathematical results that they discovered. Pythagoras is remembered today for Pythagoras’s Theorem, which states that for a right-angled triangle that the square of the hypotenuse is equal to the sum of the square of the other two sides. The Pythagorean’s discovered the irrationality of the square root of two and as this result conflicted in a fundamental way with their philosophy that number is everything, and they suppressed the truth of this mathematical result.
- 4.
Euclid was a third century B.C. Hellenistic mathematician and is considered the father of geometry.
- 5.
Euler was an eighteenth century Swiss mathematician who made important contributions to mathematics and physics. His contributions include graph theory (e.g., the well-known formula V − E + F = 2), calculus, infinite series, the exponential function for complex numbers, and the totient function.
- 6.
Other bases have been employed such as the segadecimal (or base-60) system employed by the Babylonians. The decimal system was developed by Indian and Arabic mathematicians between 800–900AD, and it was introduced to Europe in the late twelfth/early thirteenth century. It is known as the Hindu–Arabic system.
- 7.
Wilhelm Gottfried Leibniz was a German philosopher, mathematician and inventor in the field of mechanical calculators. He developed the binary number system used in digital computers, and invented the Calculus independently of Sir Issac Newton. He was embroiled in a bitter dispute towards the end of his life with Newton, as to who developed the calculus first.
References
Explication de l’Arithmétique Binaire Wilhelm Gottfried Leibniz. Memoires de l’Academie Royale des Sciences. 1703.
A Symbolic Analysis of Relay and Switching Circuits. Claude Shannon. Masters Thesis. Massachusetts Institute of Technology. 1937.
Number Theory for Computing. Song Y. Yan 2nd Edition. Springer. 1998.
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O’Regan, G. (2016). Number Theory. In: Guide to Discrete Mathematics. Texts in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-44561-8_3
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