Exercises 3 You should attempt the following questions for the next tutorial (1) For each of the following pairs of numbers a and b compute their gcd and then find integers x and y such that gcd(a, b) = ax + by. Carry out these calculations first using the extended Euclidean algorithm and then using Blankinship’s algorithm. (a) 741,715. (b) 2072,1813. (c) 1485, 1745. (2) Prove Binet’s formula for the nth Fibonacci number fn by induction. (3) Prove by induction that fn > φn−2 for n ≥ 3. (4) Use L’Hôpital’s rule to prove that li(x) = 1. x→∞ x/ ln(x) lim (5) Use the Prime Number theorem to estimate the number of primes with 300 digits. (6) Factorize 13837 using Fermat’s factorization method. (7) Factorize 809009 using Fermat’s factorization method. (8) Factorize the number 109 − 39 into primes. Homework question Read and understand Zagier’s proof that every prime that leaves remainder 1 when divided by 4 can be written as a sum of two squares. You will have to look things up. I will ask to see your attempts either individually or in groups in a couple of weeks. These are further questions for practice and background (1) Determine which of the following linear Diophantine equations has a solution and, for those that do, find all solutions. (a) 1492x + 1066y = −4. (b) 1485x + 1745y = 15. (c) 114x + 312y = 28. (2) Find the number of positive solutions of the linear Diophantine equation 10x + 28y = 1240. 1 2 (3) Factorize the number 109 + 39 into primes. (4) Define σ(n) to be the sum of all divisors of n. We say that a number n is perfect if σ(n) = 2n. This is equivalent to saying that the sum of the proper divisors of n is n. Thus both 6 and 28 are perfect. (a) Show that σ is a multiplicative function meaning that if gcd(a, b) = 1 then σ(ab) = σ(a)σ(b). e+1 (b) Show that if pe is a prime power then σ(pe ) = p p−1−1 . Thus σ(p) = p + 1 and σ(2e ) = 2e+1 − 1. (c) Prove that if p is a prime and 2p −1 is a prime then 2p−1 (2p − 1) is perfect. [Hint: gcd(2p−1 , 2p − 1) = 1]. 5 (5) The fifth Fermat number is F5 = 22 + 1. Observe that 641 = 54 + 24 = 5 · 27 + 1. Show that 641 | 54 · 228 + 232 and that 641 | 54 · 228 − 1. Deduce that F5 is not a prime. (6) Remember that Fn is the nth Fermat prime. Prove that if m < n then gcd(Fm , Fn ) = 1 by showing that Fn − 2 is a multiple of Fm . Use this result to find another proof that there are infinitely many primes. [Hint: think about the smallest prime divisors of the Fermat numbers]. (7) Prove that there are infinitely many primes of the form 4q + 3. Hint: use the idea behind Euclid’s proof that there are infinitely many primes. Let p1 , . . . , pk be all the primes of this type. Form the number N = 4p1 . . . pk − 1. Show that the product of two primes of the form 4s + 1 is again of this form. Deduce that N must therefore have a prime divisor of the form 4t + 3. [Aside: there are also infinitely many primes of the form 4q + 1 but the proof is, surprisingly, much harder]. (8) Prove that n 1 1 fn+1 fn = 1 0 fn fn−1 By taking determinants deduce that (−1)n = fn+1 fn−1 − fn2 . This identity is known as Cassini’s identity. What information does it contain? 3 (9) Consider the following sequence of numbers r q q √ √ √ 1, 1 + 1, 1 + 1 + 1, 1 + 1 + 1 + 1, . . . In √ other words, start with s1 = 1 and then compute sn+1 = 1 + sn for n ≥ 2. What do you observe? Can you prove this? (10) Below is a picture of a regular pentagon. I have assumed that the length of the sides is 1. Prove that the length of a diagonal, such as BE, is equal to φ. You may use the following result called Ptolomy’s Theorem. Ptolomy was an influential Greek astronomer who lived in the second century AD. To state his theorem I need to refer to the diagram below: 4 A cyclic quadrilateral is one whose vertices lie on the circumference of a circle. The quadrilateral ABCD above is cyclic. The theorem states that AC · BD = AD · BC + CD · AB, where AC means the length of the line, and so on.