MATH 30200 - number theory Homework set 3, due March 19, 2020 before 4 pm Exercise 1: We take, uniformly at random, an odd integer between 3 and 1010000. We use Rabin-Miller test with 200 independent random bases, and the odd integer always passes the test. Show that it is prime with probability 1 - (1/K), where K ? 10100. . Exercise 2: Let p be a prime number larger than or equal to 7. Show the following statements: - 2 is a quadratic residue of p if and only if p is congruent to 1 or 3 modulo 8, 3 is a quadratic residue of p if and only if p is congruent to 1 or 11 modulo 12, 5 is a quadratic residue of p if and only if p is congruent to 1 or 4 modulo 5. Exercise 3: Determine if 101, 102, 103, 104 are or not quadratic residues modulo 2017. Exercise 4: Let p be an odd prime number. Under which condition of congruence satisfied by p the equation x2 - x - 1 = 0 admits two solutions x in Z/pZ? Under which condition on p the equation x2 - x - 1 = 0 admits exactly one solution x in Z/pZ? Exercise 5: Let p be a prime number congruent to 3 modulo 4, and let a E Z/pZ. We assume that a is a quadratic residue modulo p. Compute a(p+1)/2 in Z/pZ, and de- duce an integer k, explicitly written in function of p, such that ak is a square root of a in Z/pZ. 1
MATH 30200 - number theory Homework set 4, due April 23, 2020 before 4 pm Exercise 1: Give a parametrization of the set of all solutions (x,y) I Z2 of the Diophan- tine equation 53x + 37y = 100. Exercise 2: Let a and b be strictly positive coprime integers. (1) Find a particular solution, and then all the solutions (u,v) I Z, of the Diophan- tine equation au+ bv = ab- a - b. Deduce that no solution has both u and v nonnegative. (2) For c > ab- a - b, we consider the solution of au+ bv= c, such that u takes the smallest possible nonnegative value. Show that usb and deduce that v ? 0. (3) Show that ab- a - b is the smallest integer which cannot be written as au+ bv for u and v nonnegative integers. Exercise 3: We assume that (x,y,2016) is a Pythagorean triple. (1) Let d be the greatest common divisor of x,y,2016. By using the general formula for the Pythagorean triples, show that d is divisible by 32. (2) Why -1 is not a quadratic residue modulo 7? Deduce that for integers u and v, u2 + v2 is divisible by 7 only when u and v are divisible by 7. (3) Deduce that d is divisible by 7. (4) Prove that d is divisible by 9. (5) 5) Deduce that there is no Pythagorean