Math 312, Lecture 1. Zinovy Reichstein. September 9, 2015 Math 312
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1 Math 312, Lecture 1 Zinovy Reichstein September 9, 2015 Math 312
2 Number theory Number theory is a branch of mathematics
3 Number theory Number theory is a branch of mathematics which studies the properties of integers (or whole numbers).
4 Number theory Number theory is a branch of mathematics which studies the properties of integers (or whole numbers). Its origins go back to (at least) 4000 years, to ancient Baylon.
5 Number theory Number theory is a branch of mathematics which studies the properties of integers (or whole numbers). Its origins go back to (at least) 4000 years, to ancient Baylon. In particular, the ancient Babylonians knew of what we now call Pythagorean triples, (a, b, c) of positive integers such that a 2 + b 2 = c 2.
6 Number theory Number theory is a branch of mathematics which studies the properties of integers (or whole numbers). Its origins go back to (at least) 4000 years, to ancient Baylon. In particular, the ancient Babylonians knew of what we now call Pythagorean triples, (a, b, c) of positive integers such that a 2 + b 2 = c 2. Number theory flourished in ancient Greece (as did other areas of mathematics).
7 Number theory Number theory is a branch of mathematics which studies the properties of integers (or whole numbers). Its origins go back to (at least) 4000 years, to ancient Baylon. In particular, the ancient Babylonians knew of what we now call Pythagorean triples, (a, b, c) of positive integers such that a 2 + b 2 = c 2. Number theory flourished in ancient Greece (as did other areas of mathematics). Some of the foundaional concepts still bear the name of Archimedes, Diophantus, Euclid, Pythagoras.
8 Number theory Number theory is a branch of mathematics which studies the properties of integers (or whole numbers). Its origins go back to (at least) 4000 years, to ancient Baylon. In particular, the ancient Babylonians knew of what we now call Pythagorean triples, (a, b, c) of positive integers such that a 2 + b 2 = c 2. Number theory flourished in ancient Greece (as did other areas of mathematics). Some of the foundaional concepts still bear the name of Archimedes, Diophantus, Euclid, Pythagoras. The discoveries of ancient greek mathematicians were lost in Europe during the dark ages, over a period of several centuries.
9 Number theory Number theory is a branch of mathematics which studies the properties of integers (or whole numbers). Its origins go back to (at least) 4000 years, to ancient Baylon. In particular, the ancient Babylonians knew of what we now call Pythagorean triples, (a, b, c) of positive integers such that a 2 + b 2 = c 2. Number theory flourished in ancient Greece (as did other areas of mathematics). Some of the foundaional concepts still bear the name of Archimedes, Diophantus, Euclid, Pythagoras. The discoveries of ancient greek mathematicians were lost in Europe during the dark ages, over a period of several centuries. During this period number theory continued to be advanced in China, India and the Middle East. The concept of zero and positional representation of numbers were known to Indian mathematicians by the 7th century. Important advances in algebra were made by Persian and Arab mathematicians throughout this period.
10 Fermat andd Euler
11 Fermat andd Euler During the Renaissance classical number theory returned to Europe.
12 Fermat andd Euler During the Renaissance classical number theory returned to Europe. Most of the theoretical material in this course is based on the work of great number theorists of the 17th and the 18th centuries,
13 Fermat andd Euler During the Renaissance classical number theory returned to Europe. Most of the theoretical material in this course is based on the work of great number theorists of the 17th and the 18th centuries, Pierre de Fermat ( ) and Leonhard Euler ( ).
14 Pierre de Fermat ( ), first photo
15 Pierre de Fermat ( ), first photo Fermat was a prominent French lawer, whose work combined classical number theory with newly developed algebraic methods.
16 Pierre de Fermat ( ), second photo
17 Pierre de Fermat ( ), second photo Fermat s Last Theorem : Let n 3 be an integer. Then x n + y n z n for any triple of positive integers x, y, z.
18 Leonhard Euler ( )
19 Leonhard Euler ( ) Leonhard Euler is considered to be the preeminent 18th century mathematician and one of the greatest mathematicians in history. He worked in a range of subjects, including number theory, graph theory and calculus.
20 Leonhard Euler ( ), second photo
21 Leonhard Euler ( ), second photo Leonhard Euler was Swiss,
22 Leonhard Euler ( ), second photo Leonhard Euler was Swiss, but he spent most of his life in Germany (Prussia) and Russia (St. Petersburg).
23 Leonhard Euler ( ), second photo Leonhard Euler was Swiss, but he spent most of his life in Germany (Prussia) and Russia (St. Petersburg). In addition in mathematics, he was also renowned
24 Leonhard Euler ( ), second photo Leonhard Euler was Swiss, but he spent most of his life in Germany (Prussia) and Russia (St. Petersburg). In addition in mathematics, he was also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.
25 Applications of number theory Number theory was long viewed as The queen of mathematics,
26 Applications of number theory Number theory was long viewed as The queen of mathematics, an area of pure research, akin to painting or poetry,
27 Applications of number theory Number theory was long viewed as The queen of mathematics, an area of pure research, akin to painting or poetry, valued for its intrinsic beauty, not applicability.
28 Applications of number theory Number theory was long viewed as The queen of mathematics, an area of pure research, akin to painting or poetry, valued for its intrinsic beauty, not applicability. This view was most directly expressed in the 1940 essay A mathematician s apology
29 Applications of number theory Number theory was long viewed as The queen of mathematics, an area of pure research, akin to painting or poetry, valued for its intrinsic beauty, not applicability. This view was most directly expressed in the 1940 essay A mathematician s apology by the British mathematician G.H. Hardy.
30 Applications of number theory Number theory was long viewed as The queen of mathematics, an area of pure research, akin to painting or poetry, valued for its intrinsic beauty, not applicability. This view was most directly expressed in the 1940 essay A mathematician s apology by the British mathematician G.H. Hardy. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity,
31 Applications of number theory Number theory was long viewed as The queen of mathematics, an area of pure research, akin to painting or poetry, valued for its intrinsic beauty, not applicability. This view was most directly expressed in the 1940 essay A mathematician s apology by the British mathematician G.H. Hardy. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.
32 Applications of number theory Number theory was long viewed as The queen of mathematics, an area of pure research, akin to painting or poetry, valued for its intrinsic beauty, not applicability. This view was most directly expressed in the 1940 essay A mathematician s apology by the British mathematician G.H. Hardy. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years. This turned out to be spectacularly false. Number theory turned out to have far-reaching applications in the computer age, both industrial and military.
33 Applications of number theory Number theory was long viewed as The queen of mathematics, an area of pure research, akin to painting or poetry, valued for its intrinsic beauty, not applicability. This view was most directly expressed in the 1940 essay A mathematician s apology by the British mathematician G.H. Hardy. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years. This turned out to be spectacularly false. Number theory turned out to have far-reaching applications in the computer age, both industrial and military. Applications, to cryptography, will be covered in the course.
34 The Well-Ordering Principle Every subset of the positive integers has a least (i.e., the smallest) element.
35 The Well-Ordering Principle Every subset of the positive integers has a least (i.e., the smallest) element. For us this will be one of the axioms of the natural numbers.
36 The Well-Ordering Principle Every subset of the positive integers has a least (i.e., the smallest) element. For us this will be one of the axioms of the natural numbers. Note that the well-ordering principle is false for other number systems, such as all integers or the positive real numbers.
37 Mathematical induction Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains n then S contains n + 1.
38 Mathematical induction Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains n then S contains n + 1. Then S contains every positive integer, i.e., S = N.
39 Mathematical induction Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains n then S contains n + 1. Then S contains every positive integer, i.e., S = N. In practice we use this to prove that some property P n is satisfied by every positive integer n as follows.
40 Mathematical induction Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains n then S contains n + 1. Then S contains every positive integer, i.e., S = N. In practice we use this to prove that some property P n is satisfied by every positive integer n as follows. (i) First we prove that P 1 is satisfied. This is called the base case.
41 Mathematical induction Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains n then S contains n + 1. Then S contains every positive integer, i.e., S = N. In practice we use this to prove that some property P n is satisfied by every positive integer n as follows. (i) First we prove that P 1 is satisfied. This is called the base case. (ii) Then we prove that if P n is satisfied, then P n+1 is satisfied. This is called the induction step.
42 Mathematical induction Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains n then S contains n + 1. Then S contains every positive integer, i.e., S = N. In practice we use this to prove that some property P n is satisfied by every positive integer n as follows. (i) First we prove that P 1 is satisfied. This is called the base case. (ii) Then we prove that if P n is satisfied, then P n+1 is satisfied. This is called the induction step. If we can establish (i) and (ii), then property P n will be true for every integer n 1. To see that this proof method is valid, denote the set of integers n such that P n is satisfied by S, and use the principle of mathematical induction to show that S = N.
43 Mathematical induction examples/exercises 1. Show that (2n 1) = n Show that 1 + q + q q n = qn+1 1 q 1 3. Show that 2 n > n 2 for any n 5. for any real number q Show that n lines in general position subdivide the plane into n(n + 1) + 1 regions Show that n 3 n is divisible by 3 for any n 1.
44 Strong mathematical induction The following variant on the principle of mathematical induction is often convenient.
45 Strong mathematical induction The following variant on the principle of mathematical induction is often convenient. Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains every integer n then S contains n + 1.
46 Strong mathematical induction The following variant on the principle of mathematical induction is often convenient. Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains every integer n then S contains n + 1. Then S contains every positive integer, i.e., S = N.
47 Strong mathematical induction The following variant on the principle of mathematical induction is often convenient. Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains every integer n then S contains n + 1. Then S contains every positive integer, i.e., S = N. In practice we use this to prove that some property P n is satisfied by every positive integer n as follows.
48 Strong mathematical induction The following variant on the principle of mathematical induction is often convenient. Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains every integer n then S contains n + 1. Then S contains every positive integer, i.e., S = N. In practice we use this to prove that some property P n is satisfied by every positive integer n as follows. Base case: First we prove that P 1 is satisfied.
49 Strong mathematical induction The following variant on the principle of mathematical induction is often convenient. Suppose a set S of positive integers (i) contains 1, and (ii) has the property that if S contains every integer n then S contains n + 1. Then S contains every positive integer, i.e., S = N. In practice we use this to prove that some property P n is satisfied by every positive integer n as follows. Base case: First we prove that P 1 is satisfied. Induction step: Then we prove that if P 1,..., P n are all satisfied, then P n+1 is satisfied as well.
50 Strong mathematical induction exercises 1. Show that every integer n 2 is either a prime or a product of two or more primes. 2. Any integer amount of postage of 12 cents or more, can be paid using only 3-cent and 5-cent stamps. 3. The nth Fibonacci number a n is defined by the recursive formula a 1 = a 2 = 1, a n+2 = a n+1 + a n. Show that for any n 1. Here a n = 1 5 (α n β n ) α = and β =
51 Strong mathematical induction exercises 1. Show that every integer n 2 is either a prime or a product of two or more primes. 2. Any integer amount of postage of 12 cents or more, can be paid using only 3-cent and 5-cent stamps. 3. The nth Fibonacci number a n is defined by the recursive formula a 1 = a 2 = 1, a n+2 = a n+1 + a n. Show that for any n 1. Here a n = 1 5 (α n β n ) α = and β = Note that α and β are the roots of the quadratic equation x 2 x 1 = 0.
52 Strong mathematical induction exercises 1. Show that every integer n 2 is either a prime or a product of two or more primes. 2. Any integer amount of postage of 12 cents or more, can be paid using only 3-cent and 5-cent stamps. 3. The nth Fibonacci number a n is defined by the recursive formula a 1 = a 2 = 1, a n+2 = a n+1 + a n. Show that for any n 1. Here a n = 1 5 (α n β n ) α = and β = Note that α and β are the roots of the quadratic equation x 2 x 1 = 0. That is, α 2 = α + 1 and β 2 = β + 1.
53 Remarks Remark 1. Mathematical induction can be used to prove that property P n
54 Remarks Remark 1. Mathematical induction can be used to prove that property P n is valid for every integer n d, where d is not necessarily 1.
55 Remarks Remark 1. Mathematical induction can be used to prove that property P n is valid for every integer n d, where d is not necessarily 1. For example, 2 n > n 2 for every n 5.
56 Remarks Remark 1. Mathematical induction can be used to prove that property P n is valid for every integer n d, where d is not necessarily 1. For example, 2 n > n 2 for every n 5. Here we use the substitution m = n d + 1 and argue by induction with respect to m. Note that m 1 corresponds to n d. Remark 2. The well-ordering principle, the principle of mathematical induction
57 Remarks Remark 1. Mathematical induction can be used to prove that property P n is valid for every integer n d, where d is not necessarily 1. For example, 2 n > n 2 for every n 5. Here we use the substitution m = n d + 1 and argue by induction with respect to m. Note that m 1 corresponds to n d. Remark 2. The well-ordering principle, the principle of mathematical induction and the principle of strong mathematical induction
58 Remarks Remark 1. Mathematical induction can be used to prove that property P n is valid for every integer n d, where d is not necessarily 1. For example, 2 n > n 2 for every n 5. Here we use the substitution m = n d + 1 and argue by induction with respect to m. Note that m 1 corresponds to n d. Remark 2. The well-ordering principle, the principle of mathematical induction and the principle of strong mathematical induction are all equivalent to each other.
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