What is number theory? Briefly, it is the study of the integers and questions arising from them.<\/p>\n
Definition 1.0.1 The Integers.\u00a0<\/b>The set of\u00a0counting numbers<\/a>\u00a0<\/b>is denoted<\/p>\n N =\u00a0{<\/i>0,\u00a0<\/i>1,\u00a0<\/i>2,\u00a0<\/i>3,\u00a0<\/i>4, \u00b7 \u00b7 \u00b7 }<\/i>.<\/p>\n Note that in this text, this set begins at zero1. The\u00a0integers\u00a0<\/b><\/a>is the set of positive\u00a0and\u00a0<\/i>negative counting numbers<\/a>:<\/p>\n Z =\u00a0{\u00b7 \u00b7 \u00b7 ,\u2212<\/i>3,\u2212<\/i>2,\u2212<\/i>1,\u00a0<\/i>0,\u00a0<\/i>1,\u00a0<\/i>2,\u00a0<\/i>3, \u00b7 \u00b7 \u00b7 }<\/i>.<\/p>\n This is a fairly dry definition, though. The best way to find out what this definition means is to try to answer some questions\u00a0about\u00a0<\/i>integers!<\/p>\n 1.1 A First Problem<\/b><\/p>\n Let\u2019s start! Suppose you have lots of left-over postage stamps2 that are of just a few different denominations. It could be fun to see what amounts you could make from them.<\/p>\n To be concrete, let\u2019s assume first that all your stamps are numbered 2\u00a2<\/a> and 3\u00a2. Here are two questions we could ask. They are mathematically equivalent, but might take your exploration in two very different directions!<\/p>\n