One such process is factoring: it is easy for a computer to multiply two large prime numbers together, but very difficult to recover the prime numbers given their product. The two most common such processes both come from number theory. Many cryptosystems require a computationally difficult one-way process, which is quick to do but hard to reverse. Some basic problems in elementary number theory are well-suited for use in modern cryptography. Some examples of the latter will be outlined in the following section. These theorems have a myriad of diverse applications to, for instance, primality testing, periods of decimal expansions of fractions, and modern cryptography.
If \(n\ge 3\) is a positive integer, can a positive \(n^\text \equiv 1 \pmod n.\) Which integers can be written as the sum of four squares?
There are an abundance of simply formulated questions about the integers that involve little more than the basics of addition and multiplication (the ring operations on the integers), but which are nevertheless unsolved or extremely difficult to solve. "Mathematics is the queen of the sciences, and number theory is the queen of mathematics." Because of the fundamental nature of the integers in mathematics, and the fundamental nature of mathematics in science, the famous mathematician and physicist Gauss wrote: Of particular importance in this text is the author's emphasis on the value of numerical examples in number theo.Number theory is the study of properties of the integers. Later chapters offer lucid treatments of quadratic congruences, additivity (including partition theory) and geometric number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems.Īmong the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers.
In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.
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