Math 250 Take-Home Final Exam

Math 250 Take-Home Final Exam – Due Thursday, May 14 at Noon There are seven questions for a total of 100 points plus 10 points of extra credit. 1. (15+5 points) (See Exercise 27.2, pp. 209-210.) Liouville’s lambda function λ(n) is defined by factoring n into a product of primes, n = p1 k1 p2 k2 · · · pt kt , and then setting λ(n) = (−1)k1 +k2 +···kt . (Also, we let λ(1) = 1.) (a) Compute the following values of Liouville’s function: λ(30); λ(504); λ(60750). (b) Prove that Liouville’s function λ is multiplicative: i.e., prove that if gcd(m, n) = 1, then λ(mn) = λ(m)λ(n). (c) Define a new function G(n) by the formula X G(n) = λ(d) = λ(d1 )+λ(d2 )+· · ·+λ(dr ), where d1 , d2 , . . . , dr are the divisors of n. d|n Compute the value of G(n) for all 1 ≤ n ≤ 18. (d) Use your computations in (c), to make a conjecture as to the value of G(n). (e) (Extra credit 5 pts.) Prove that you conjecture in (d) is correct. 2. (15 points) (See Exercise 30.1, p. 234, and the hint there.) Show that the equation y 2 = x3 + xz 4 has no solutions in nonzero integers. 3. (10 points) (Based on Exercise 41.5, pp. 363-364.) Consider the group of rational points E(Q) on the elliptic curve E defined by the equation E : y 2 = x3 − 3x + 7. (a) Let P = (2, 3) ∈ E(Q). Compute the rational 2P ∈ E(Q). (b) Compute 3P ∈ E(Q). 4. (15 points) (See Exercise 42.4, p. 371.) Consider the elliptic curve E defined over Q by the equation E : y 2 = x3 + 7. In this problem, we’ll show that E(Q) contains no points (x, y) with integer coordinates. (a) Suppose (x, y) ∈ E(Q) with x, y ∈ Z. Show that x must be odd. (b) Show that y 2 + 1 = (x + 2)(x2 − 2x + 4). (c) Show (x2 − 2x + 4) ≡ 3 (mod 4). Deduce that (x2 − 2x + 4) must be divisible by some prime q ≡ 3 (mod 4). (d) Reduce the original equation for E modulo q and use the result to show that −1 must be square (mod q). Explain why this shows E has no integer solutions. 5. (10+5 points) (See Exercise 43.1, p. 382 and the hints there.) For each prime p, let Mp be the number of solutions modulo p to the equation x2 + y 2 = 1. (a) Compute the following values of M3 , M5 , M7 , M11 , M13 and M17 . (b) Make a conjecture for the value of Mp for any prime p ≥ 3 (c) (Extra credit 5 pts.) Prove your conjecture. 6. (20 points) (See Exercise 43.5, p. 383, and the table of values for ap there.) Consider the elliptic curve E : y 2 = x3 + 1. In this problem, we’ll investigate patterns in the values of ap and prove a conjecture. (a) Make a conjecture as to which primes p have p-defect ap = 0. Prove your conjecture. (b) For those primes p < 71 with ap 6= 0, use the table of values for ap on p. 383 to compute the value of 4p − ap 2 . Make a conjecture as to what is special about these numbers. (c) For every prime p < 71 with p ≡ 1 (mod 3), find all pairs of integers (A, B) such that 4p = A2 + 3B 2 . (d) Compare the values of A and B with the value of ap given in the table on p. 383. Conjecture how there are related. (e) For each of the primes p below, the pairs (A, B) such that 4p = A2 + 3B 2 are given. Use your conjecture in (d) to guess the value of ap for these primes p. p = 541 (A, B) = (46, 4), (29, 21), (17, 25) p = 2029 (A, B) = (79, 25), (77, 27), (2, 52) p = 8623 (A, B) = (173, 39), (145, 67), (28, 106) 7. (15 points) (See Exercise 44.4(a), p. 387.) Let p be a fixed prime greater than or equal to 5. Consider the elliptic curve E : y 2 = x3 + p. (a) Show that p is a prime of bad reduction for E. (b) Determine the value of ap and prove that your formula is correct.