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1 What is Number Theory?? The study of patterns or properties exhibited by certain sets or sequences of integers (very often, restricted to positive integers). Warm-Up Exercise: Give a simple description of the following sets of positive integers. Some of these are quite subtle. • 2, 4, 6, 8, 10, 12, . . . • 1, 3, 6, 10, 15, 21, 28, . . . • 1, 4, 7, 10, 13, . . . • 1, 2, 4, 5, 8, 9, 10, 13, . . . • 2, 3, 5, 7, 11, 13, 17, 19, . . . • 6, 28, 496, 32512, 67100672, . . . (?) • 1, 4, 9, 16, 25, 36, . . . • 5, 17, 37, 101, 197, 257, 401, . . . (?) 2 Some Common Themes Given a description for a set of integers, we may ask the following: • Does any integer satisfy this description? • Given an integer, does it belong to our set? • Is our set infinite? • What proportion of integers in a given range (often 1 ≤ x ≤ N ) belong to our set? The answers can be quite difficult! 3 Quick Examples Consider the following problems involving real numbers: • Find an ordered pair (x, y) such that 37x − 91y = 15. • Find all ordered pairs (x, y) such that x2 + y 2 = 63529. • Are there any ordered pairs (x, y) such that x2 − 13y 2 = 1? If we ask the same questions, but require x and y to be integers, these questions become much more interesting. 4 Prime Numbers You’ve probably been taught an integer n > 1 is prime if it has exactly two positive factors: 1 and n. Otherwise, n is composite. • A more accurate term is “irreducible,” because of certain number systems considered in abstract algebra. We will continue to use “prime” here. • n = 1 IS NEITHER PRIME NOR COMPOSITE (there are very good reasons for doing this). • There are infinitely many prime integers (this was known to the ancient Greeks; we’ll see a proof due to Euclid later in the course). • But in general, it can be difficult to determine whether a given integer is prime. Is n = 418, 917, 568, 138, 419, 379 prime or composite? 5 Why are Primes Important? In basic arithmetic: Every integer greater than 1 can be written as a product of prime numbers in an “essentially unique” way. This result is known as the Fundamental Theorem of Arithmetic. • Quick example: Find the prime factorization of 360. To get uniqueness, write the prime factors in nondecreasing order (you may wish to write repeated factors using exponents, if you find that more familiar). • Not-so-quick example: 292, 460, 418, 515, 073, 979 is the product of two primes. Can you find them? (this requires more sophisticated techniques or the aid of a computer, so don’t try too hard). VERY LARGE primes are essential to secure communication over an insecure medium (such as the Internet). 6 Distribution of Primes For a given x > 0, how many primes are less than or equal to x? • Answer: Approximately x/ ln(x). More precisely, if π(x) denotes the number of primes less than or equal to x, then π(x) = 1. x→+∞ x/ ln(x) lim • This result (conjectured in the late 1700’s, but not proven in 1896) is known as The Prime Number Theorem. There are several known proofs, but they are quite complicated and require material beyond the scope of this course. 7 Primes of a Given Form • Except for n = 2, every prime can be written as either 4k + 1 or 4k + 3 for some integer k. How many primes of each kind? – 4k + 1 primes: 5, 13, 17, 29, 37, 41, . . . – 4k + 3 primes: 3, 7, 11, 19, 23, 31, . . . • Question: Are there infinitely many primes of both kinds? – Answer: Yes! We’ll prove both using an argument similar to Euclid (the 4k + 3 case is easy, the 4k + 1 case requires more work). • For large enough x, roughly half of all primes less than x have the 4k + 1 form, and roughly half have the 4k + 3 form (this is difficult to prove). 8 Primes in Arithmetic Progressions Theorem (Dirichlet 1837): Let a and q > 1 be integers whose greatest common factor is 1. There are infinitely many primes of the form qk + a. • The previous slide has q = 4, with a = 1 or a = 3 (Question: Why does this essentially cover all posible values of a?) • Take q = 5 as another example. There are inifinitely many primes of each of the following forms: 5k + 1, 5k + 2, 5k + 3, and 5k + 4. • It also turns out that the primes are “uniformly distributed” among all possible forms for a given value of q (up to any given value of x, there are roughly the same number of primes of each form). • The general case (and uniform distribution) require material beyond the scope of this course, but we’ll prove some special cases later on. 9 Some Deceptively Simple Questions • For all n, is there a prime between n2 and (n + 1)2? – √ Since the number of perfect squares smaller than x is approximately x, the Prime Number Theorem says there should be many more prime numbers than perfect squares. – A. Ingham (1937): For large enough n, there is always a prime between n3 and (n + 1)3. • Are there infinitely many primes of the form n2 + 1? – H. Iwaniec (1978): There are infinitely many numbers n2 + 1 that are either prime or the product of two primes. 10 Unique Prime Factorization is not universal! • Let E be the set of even positive integers (Abstract Algebra: if negative evens are included, E is a ring). We’ll say that an element of E is an E-prime if it cannot be written as the product of two even integers. • The first four E-primes are 2, 6, 10, 14. List the next four. Can you give a description of all E-primes? • E-Prime Factorization is not necessarily unique. Try the following: – Write 60 as a product of E-primes in two different ways (writing the E-prime factors in nondecreasing order). – Find an integer with at least three different E-prime factorizations. 11 Other Representation Problems • Any positive integer n has a unique base-10 representation: n = (a0) + 10 · (a1) + 102 · (a2) + · · · + 10k · (ak ), where each ai is an integer, 0 ≤ ai < 10, and ak 6= 0. You’ve used this representation for a long time; it’s what allows you to do basic arithmetic. • This can be done with powers of any integer base greater than 1 . • In particular, base 2 (binary) is IMPORTANT in Computer Science: every positive integer can be written uniquely as a sum of powers of 2. 12 Sums of Squares • Which positive integers n can be written as n = x2 + y 2, where x and y are both integers? The cases n = 1 and n = 2 are very simple. • However, this can’t √ be done for n = 3. Both x and y would have to be integers less than 3. If you try all possibilities, none will work. • n = 63529 (from earlier) can be written in this way. (x, y) = (5, 252) is one possibility; there are 31 others. • The positive integers that can be written as a sum of two squares start with 1, 2, 4, 5, 8, 9, 10, 13, . . . . There is a very interesting (and not obvious) pattern that emerges. Try it for prime n if you’re curious. 13 More Representation Problems • What about sums of three squares? (n = 7 is the smallest integer that cannot be written in this way). • Sums of four or more squares? (try it for yourself). • There’s nothing special about sums of squares. We could try sums of cubes, fourth powers, etc. • Waring’s Problem (1770): Given an integer k ≥ 2, how many terms are needed to represent every positive integer as a sum of perfect kth powers? (There has been a great deal of successful work on this problem, but a complete solution is still unknown). 14 Goldbach’s Conjecture (1742) Is every even number bigger than two a sum of two primes? • 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, 10 = 7 + 3, etc. • Computer searches have verified this for every n < 1018. • J. Chen (1966): Every even n ≥ 4 can be expressed as a sum of a prime and a “P2” (a prime or a product of two different primes). • O. Ramare (1995): Every even n ≥ 4 is the sum of at most 6 primes. 15 A Well-Known Equation • Can you find solutions to x2 + y 2 = z 2 if each variable is required to be a nonzero integer (Question: why would we exclude zero?). • (x, y, z) is called a Pythagorean Triple if it satisfies the above conditions. (3, 4, 5) and (5, 12, 13) are two examples. There are infinitely many. • What about x3 + y 3 = z 3? What about x4 + y 4 = z 4? • What about xn + y n = z n, for some other choice of exponent n? 16 Congruence Arithmetic For integers a and b, the notation a ≡ b (mod n) means that a − b is an integer multiple of n. “Clock arithmetic” uses n = 12. • For a fixed value of n, a ≡ b (mod n) is an equivalence relation, which roughly means that the congruence a ≡ b (mod n) behaves similarly to the ordinary equation a = b. • Many (but not all!) familiar arithmetic and algebraic operations will also work on congruences (example: you can add/subtract the same value from both sides of a congruence). • We often get interesting/useful results if we replace an ordinary equation by a congruence. Some examples on the next slide... 17 If we require x to be an integer, the following equations have no solutions: • 4x = 2 • x2 + 1 = 0 • x4 = 1. Exercise: Replace each equation by a congruence (mod 5). Are there any integers x satisfying the congruence? If there are any solutions, can you find them all? Repeat this using congruences (mod 6) and (mod 7). Note: We’ll study congruences in a more organized way later in the course.