an introduction to diophantine equations a problem-based approach.pdf

[...]... 239 1.7 Miscellaneous Diophantine Equations 253 II.2 Solutions to Some Classical Diophantine Equations 265 2.1 Linear Diophantine Equations 265 2.2 Pythagorean Triples and Related Problems 273 2.3 Other Remarkable Equations 278 II.3 Solutions to Pell-Type Equations 289 3.1 Solving Pell’s Equation by Elementary Methods 289 3.2 The Equation ax2 − by 2 = 1 ... integers such that xy + yz + zx − xyz = 2 3 Determine all triples (x, y, z) of positive integers such that (x + y)2 + 3x + y + 1 = z 2 (Romanian Mathematical Olympiad) 18 Part I Diophantine Equations 4 Determine all pairs (x, y) of integers that satisfy the equation (x + 1)4 − (x − 1)4 = y 3 (Australian Mathematical Olympiad) 5 Prove that all the equations x6 + ax4 + bx2 + c = y 3 , where a ∈ {3, 4, 5},... Some Classical Diophantine Equations 67 2.1 2.2 Pythagorean Triples and Related Problems 76 2.3 I.3 Linear Diophantine Equations 67 Other Remarkable Equations 88 Pell-Type Equations 117 3.1 3.2 Solving Pell’s Equation 121 3.3 The Equation ax2 − by 2 = 1 135 3.4 I.4 Pell’s Equation: History and Motivation 118 The Negative Pell’s Equation... 8}, are not solvable in positive integers (Dorin Andrica) 6 Solve in positive integers the equation x2 y + y 2 z + z 2 x = 3xyz 7 Find all integer solutions to the equation (x2 − y 2 )2 = 1 + 16y (Russian Mathematical Olympiad) 8 Find all integers a, b, c, x, y, z such that a + b + c = xyz, x + y + z = abc, and a ≥ b ≥ c ≥ 1, x ≥ y ≥ z ≥ 1 (Polish Mathematical Olympiad) 1.3 Solving Diophantine Equations. .. such that xy = y x 13 Solve in positive integers the equation xy + y = y x + x 14 Let a and b be positive integers such that ab+1 divides a2 +b2 Prove that a2 +b2 ab+1 is the square of an integer (29th IMO) 15 Find all integers n for which the equation (x + y + z)2 = nxyz is solvable in positive integers (American Mathematical Monthly, reformulation) 20 Part I Diophantine Equations 1.3 The Parametric... many situations the integral solutions to a Diophantine equation f (x1 , x2 , , xn ) = 0 can be represented in a parametric form as follows: x1 = g1 (k1 , , kl ), x2 = g2 (k1 , , kl ), , xn = gn (k1 , , kl ), where g1 , g2 , , gn are integer-valued l-variable functions and k1 , , kl ∈ Z The set of solutions to some Diophantine equations might have multiple parametric representations... Mathematical Olympiad) 3 Let p and q be distinct prime numbers Find the number of pairs of positive integers x, y that satisfy the equation p q + = 1 x y (K¨MaL) o 4 Find the positive integer solutions to the equation x3 − y 3 = xy + 61 (Russian Mathematical Olympiad) 12 Part I Diophantine Equations 5 Solve the Diophantine equation x − y 4 = 4, where x is a prime 6 Find all pairs (x, y) of integers such that... + 1 = y 4 (Romanian Mathematical Olympiad) 7 Solve the following equation in nonzero integers x, y : (x2 + y)(x + y 2 ) = (x − y)3 (16th USA Mathematical Olympiad) 8 Find all integers a, b, c with 1 < a < b < c such that the number (a − 1)(b − 1)(c − 1) is a divisor of abc − 1 (33rd IMO) 9 Find all right triangles with integer side lengths such that their areas and perimeters are equal 10 Solve the... The Negative Pell’s Equation 301 II.4 Solutions to Some Advanced Methods in Solving Diophantine Equations 309 4.2 The Ring Z[i] of Gaussian Integers 309 √ The Ring of Integers of Q[ d] 314 4.3 Quadratic Reciprocity and Diophantine Equations 322 4.4 Divisors of Certain Forms 324 4.1 References 327 Glossary 331 Index 341 Part I Diophantine Equations. .. into k integer factors a1 , a2 , , ak Each such factorization yields a system of equations ⎧ ⎪ f1 (x1 , x2 , , xn ) = a1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ f2 (x1 , x2 , , xn ) = a2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ fk (x1 , x2 , , xn ) = ak Solving all such systems gives the complete set of solutions to (1) T Andreescu et al., An Introduction to Diophantine Equations: A Problem-Based Approach, DOI 10.1007/978-0-8176-4549-6_1, . class="bi x0 y0 w0 h0" alt=""