It grew out of undergr uate courses that the author taught at harvard, uc san diego, and the university of washington. Universityofillinois,1914 thesis submittedinpartialfulfillmentoftherequirementsforthe degreeof masterofarts. Introduction to cryptography by christof paar 96,253 views. This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. There are many useful facts about congruences some of. As with so many concepts we will see, congruence is simple, perhaps familiar to you, yet enormously useful and powerful in the study of number theory. Number theory using congruence to find the remainder for large values. Pages in category theorems in number theory the following 109 pages are in this category, out of 109 total. Section 1 introduces euclids algorithm, which is used to find the hcf of two integers, and the idea of congruences, mathematical statements used to compare remainders when two integers are each divided by another integer. On principal congruences and the number of congruences of a lattice with more ideals than filters.
If n is a positive integer, we say the integers a and b are congruent modulo n, and write a. This notation, and much of the elementary theory of congruence, is due. We say that \a\ is congruent to \b\ modulo \m\ if \m \mid ab\ where \a\ and \b \ are integers, i. It grew out of undergraduate courses that the author taught at harvard, uc san diego, and the university of washington. For example, here are some problems in number theory that remain unsolved. Congruence problems of number theory in mathematical. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. There are many useful facts about congruences some of which are listed in the from elec 6. It is well kno wn from the theory of congruence varieties and. This free course, introduction to number theory, is a branch of mathematics concerned with the properties of integers. This book, which presupposes familiarity only with the most elementary concepts of arithmetic divisibility properties, greatest common divisor, etc.
This is a book about prime numbers, congruences, secret messages, and. The libretexts libraries are powered by mindtouch and are supported by the department of education open textbook pilot project, the uc davis office of the provost, the uc davis library, the california state university affordable learning solutions program, and merlot. Least residue of a big power of 7 mod 50 using congruences duration. Subsequent chapters present proofs of fermats and wilsons theorems, introduce number theoretic functions, and explore the quadratic reciprocity theorem. An introduction to the theory of numbers open textbook. From the division algorithm, q and r are uniquely determined.
We also acknowledge previous national science foundation support under grant numbers 1246120, 1525057, and 14739. Congruence is an important concept in number theory. The systematic study of number theory was initiated around 300b. Congruences satisfy a number of important properties, and are extremely useful in many areas of number theory. Introduction to congruences mathematics libretexts.