What Is Number Theory? - Department of Mathematics

Chapter 1 What Is Number theory ? Number theory is the study of the set of positive whole numbers1;2;3;4;5;6;7; : : : ;which are often called the set ofnatural numbers. We will especially want to studytherelationshipsbetween different sorts of numbers. Since ancient times, peoplehave separated the natural numbers into a variety of different types. Here are somefamiliar and not-so-familiar examples:odd1;3;5;7;9;11; : : :even2;4;6;8;10; : : :square1;4;9;16;25;36; : : :cube1;8;27;64;125; : : :prime2;3;5;7;11;13;17;19;23;29;31; : : :composite4;6;8;9;10;12;14;15;16; : : :1 (modulo 4)1;5;9;13;17;21;25; : : :3 (modulo 4)3;7;11;15;19;23;27; : : :triangular1;3;6;10;15;21; : : :perfect6;28;496; : : :Fibonacci1;1;2;3;5;8;13;21; : : :Many of these types of numbers are undoubtedly already known to you. Oth-ers, such as the modulo 4 numbers, may not be familiar. A Number is said to becongruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4, and sim-ilarly for the 3 (modulo 4) numbers.

[Chap. 1] What Is Number Theory? 7 original number. Thus, the numbers dividing 6 are 1, 2, and 3, and 1+2+3 = 6. Similarly, the divisors of 28 are 1, 2, 4, 7, and 14, and

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