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1. Find all solutions in N of the following Diophantine equation - y²)= - y3 = 0 (Hint: divide by =³ and look for rational solutions of the equation A² - B2 - B³ = 0. This is the equation of a curve called node and it looks like an a. The point (A,B) = (0,0) is a singular point, that is any line through this point will meet the curve twice in (0,0). Use Diophantus chord method using the lines passing through (0,0).) 2. Let P be a prime. Prove or disprove the following statement: the inverse mod P of a primitive root of pis a primitive root of p. 3. = for some N € N? Is 114243 + 80282.2 = 3+2v2 for some n € Z2 (Hint: use the structure of the solutions of the Pell's equation 2 - 2y = 1.) 4. Let P be a prime. Use inverses mod P to prove that, if plab, then pla or plb. (Hint: assume y ja and use inverses mod P to find b in the relation ab III 0 mod p.) 5. Factor each of the following Gaussian integers as. a product of Gaussian primes. (a) 5 (b) 6 (c) 7 (d) 5+7i (e) 7i (f) (1+2i)12 6. Let a1 @2, @3 and n be in N. Consider the system of congruences x=a1 (mod n) x=@2 ( mod n +1) x=a3 ( mod n + 2) (a) If possible solve the system for n = 3 and @1 = 1,42 = 9and a3 = 6. (b) If possible solve the system for n = 4 and @1 = 1,a2 = 9and @3 = 6. (c) Prove that, if n is odd, then the system has solutions for any choice of and a3- (d) Prove that, if " is even, then there is a choice of and a3 such that the system does not have solutions (Hint: consider the first and third equation.)

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