Abstract
The theory of numbers is the oldest and the most fundamental mathematical discipline. In the last 30 years it has become very important, mostly due to applications to cryptography. In this chapter we familiarise the reader with the basics of number theory necessary to understand the RSA cryptosystem that will appear in Chap. 2. We make emphasis on the prime factorisation, the greatest common divisor, modular arithmetic, Euler’s function and Euler’s theorem. We state without a proof the prime number theorem and discuss the frequency of prime numbers in the number system which will further be needed for the discussion of the complexity of algorithms for finding the prime factorisation of an integer. We introduce the number-theoretic section of GAP computational package. The theory is supplemented with a number of exercises.
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Notes
- 1.
Euclid of Alexandria (about 325 BC–265 BC) is one of the most prominent educators of all times. He is best known for his treatise on mathematics The Elements which is divided into 13 books: the first six on geometry, three on number theory, one is devoted to Eudoxus’s theory of irrational numbers and the last three to solid geometry. Euclid is not known to make any original discoveries, and the elements are based on the work of the people before him such as Eudoxus, Thales, Hippocrates and Pythagoras. Over a thousand editions of this work have been published since the first printed version appeared in 1482. Very little, however, is known about his life. The enormity of work attributed to him even led some researchers to suggest that Euclid was not a historic character and that the Elements were written by a team of mathematicians at Alexandria who took the name Euclid from the historic character who lived 100 years earlier.
- 2.
Leonhard Euler (1707–1783) was a Swiss mathematician who made enormous contributions in fields as diverse as infinitesimal calculus and graph theory. He introduced much of the modern mathematical terminology and notation [3]. He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy and music theory.
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- 4.
Pafnutii Lvovich Chebyshev (1821–1894) was a Russian mathematician who is largely remembered for his investigations in number theory. Chebyshev is also famous for the orthogonal polynomials he invented. He had a strong interest in mechanics as well.
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Jacques Salomon Hadamard (1865–1963) was a French mathematician whose most important result is the prime number theorem which he proved in 1896. He worked on entire functions and zeta functions and became famous for introducing Hadamard matrices and Hadamard transforms.
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Charles Jean Gustave Nicolas, Baron de la Vallée Poussin (1866–1962) is best known for his proof of the prime number theorem and his major work Cours d’Analyse. He was additionally known for his writings about the zeta function, Lebesgue and Stieltjes integrals, conformal representation, algebraic and trigonometric series.
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Joseph Louis Francois Bertrand (1822–1900), born and died in Paris, was a professor at the École Polytechnique and Collège de France. He was a member of the Paris Academy of Sciences and was its permanent secretary for twenty-six years. Bertrand made a major contribution to group theory and published many works on differential geometry and on probability theory.
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Sun Tzu (3rd–5th century AD) (or Sun Zi) was a Chinese mathematician and astronomer. He investigated Diophantine equations. He authored “Sun Tzu’s Calculation Classic,” which contained, among other things, the Chinese remainder theorem.
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Gottfried Wilhelm von Leibniz (1646–1716) was a German mathematician and philosopher who developed infinitesimal calculus independently of Isaac Newton, and Leibniz’s mathematical notation has been widely used ever since it was published. He invented an early mechanical calculating machine.
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Slinko, A. (2020). Integers. In: Algebra for Applications. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-44074-9_1
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