MATH342 Practice Exam This exam is intended to be in a similar style to the examination in May/June 2012. It is not imlied that all questions on the real examination will follow the content of the ractice questions very closely. You should exect the unexected, esecially at the ends of exam questions, and think things out as best you can. 1. For x and y Z, and n Z + define x y mod n in terms of division by n. Prove that if gcd(n, m) = 1 then mx my mod n x y mod n. Find all integer solutions of the following. In some cases there may be no solutions. a) 3x 6 mod 9. b) 3x 5 mod 6. c) x 2 + x + 1 0 mod 5. d) x 2 1 mod 7. 3x 1 mod 7 e) Solve the simultaneous equations x 5 mod 6. 3x 2 mod 5 [20 marks] 1
2. If and q are ositive rimes with < q and n is comosite (that is, not rime) for all integers n with < n < q, then the set G(, q) = {n Z + : < n < q} is called a rime ga and its length is q. a) By considering the integers n! + k for 2 k n show that there is a rime ga of length n for each n Z +. Find the least ossible rime such that G(, q) is a rime ga of length 4. b) State the Fundamental Theorem of Arithmetic. Use it to show that, if n is comosite, then n is divisible by a rime with n. Use this to show that 89 and 97 are rime. c) Define the rime number counting function π(x). State the Prime Number Theorem. d) Let k (k 1) be an enumeration of all the ositive rimes with 1 = 2 and k < k+1 for all k 1 Prove that if n is sufficiently large, then there is a rime ga G( k, k+1 ) with k n and k+1 k > 1 ln n. 2 Hint: Let m be the largest integer with m n. Consider m k=1 ( k+1 k ) and aly the Prime Number Theorem to π(n). 3. State Fermat s Little Theorem. (i) Use it to rove that 2 99 + 3 98 is divisible by 17. (ii) Find all the ossible orders of elements of the grou of units G 17. Find all rimitive elements. Give an examle of an element of each ossible order. (iii) Let m and n Z + with n 2, and let nm 1 be divisible by 17. Show n 1 that either m is even; or m 0 mod 17 and n 1 mod 17. Find all ossible values of n in the cases when m is even but not divisible by 4, or is divisible by 4 but not divisible by 8. 2
4. Define Euler s φ function. Prove that if is a ositive rime and a Z + then φ( a ) = a 1 ( 1). Also comute the sum a (using Euler s notation) of the divisors of a. Now write down the formulas for φ(n) and n, for any n Z +, with n 2, in terms of the rime decomosition of n, where n = m i=1 k i i for distinct rimes i and integers k i 1. Comute φ(2016) and φ(11!). Find all integers n such that φ(n) = 10. 5. Let n 1 2 and n 2 2 be corime integers. Show that, for integers x and y, x y mod n 1 n 2 (x y mod n 1 x y mod n 2 ). Give an examle to show that this is not true in general if n 1 and n 2 are not corime. Recall that n Z + is a seudorime to base a if a n 1 1 mod n, and n is a Carmichael number if n is comosite and is a seudorime to base a for all a in the grou G n of units mod n. Let n = 2 11 1 = 2047. Verify that 2047 is a seudorime to base 2. Exlain why any rime dividing 2 11 1 must be 1 mod 11 and find the rime factorisation of 2047. Give Korselt s equivalent definition of a Carmichael number. Use it to verify that 2821 is a Carmichael number. Now let n be any integer 2. Show that is a subgrou of G n. {a : a n 1 1 mod n} Now let n = 35. Identify all a mod 35 such that 35 is a seudorime to base a. 3
6. In this question let Z[i] be the ring of Gaussian integers, that is Z[i] = {s + it : s, t Z} a) Show that Z[i] is a Euclidean domain with Euclidean valuation v(s + it) = s 2 + t 2. Hint: Show that if z C then there is q Z[i] such that z q 2 1 2. If a and b Z[i] with b 0, aly this with z = a/b. b) Show that if n, s, t Z and s + it divides n, then s it divides n and s 2 + t 2 divides n 2 in Z. Show also, using the fact that comlex conjugation is multilicative or otherwise, that if n 1 Z + and n 2 Z + are both the sums of the squares of two integers, then the same is true for n 1 n 2. c) Using the fact that Z[i] is a unique factorisation domain, show that if s and t are both non-zero integers and s + it is rime in Z[i], then s 2 + t 2 is rime in Z. Deduce that if s + it divides n in Z[i] then s 2 + t 2 divides n in Z. d) Exlain why any integer n which is a sum of two non-zero integer squares is the roduct of rimes with this roerty and a square of an integer. The square of the integer can be just 1 but there must be at least one rime in the roduct. e) Show that there are infinitely many rimes which are the sum of two squares. Hint: Suose there are just finitely many such rimes q i, 1 i n. Then let N 1 = n i=1 q i and N = N 2 1 + 1. Consider any rime which divides N. 4
7. Define the Legendre symbol for any ositive rime and any integer q corime to. Show that if is any odd rime then q ( 1)/2 mod, stating any theory that you use. In articular, you may assume the existence of a rimitive element in G. Deduce that F : q mod : G {±1} is a grou homomorhism. ( ) 1 State the value of for any odd rime. State Gauss Law of quadratic recirocity for for any distinct ositive rimes q and, including the case q = 2. Deduce from these laws that for any odd rime ( ) { 2 1 if = 1 or 3 mod 8 = 1 if = 1 or 3 mod 8. Comute ( ) 5 19 and ( ) 46. 89 5