By Titu Andreescu

This problem-solving booklet is an advent to the learn of Diophantine equations, a category of equations during which in simple terms integer recommendations are allowed. The presentation gains a few classical Diophantine equations, together with linear, Pythagorean, and a few better measure equations, in addition to exponential Diophantine equations. a few of the chosen routines and difficulties are unique or are provided with unique strategies. An creation to Diophantine Equations: A Problem-Based procedure is meant for undergraduates, complicated highschool scholars and academics, mathematical contest contributors ― together with Olympiad and Putnam opponents ― in addition to readers drawn to crucial arithmetic. The paintings uniquely offers unconventional and non-routine examples, principles, and strategies.

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B − 1) · A (2) modulo B. All these remainders are distinct. Indeed, if k1 A = q1 B + r and k2 A = q2 B + r for some k1 , k2 ∈ {1, 2, . . , B − 1}, then (k1 − k2 )A = (q1 − q2 )B ≡ 0 (mod B). Since gcd(A, B) = 1, it follows that |k1 − k2 | ≡ 0 (mod B). Taking into account that k1 , k2 ∈ {1, 2, . . , B − 1}, we have |k1 − k2 | < B. Thus k1 − k2 = 0. It is not diﬃcult to see that k · A ≡ 0 (mod B) for all k ∈ {1, 2, . . , B − 1}. , there exist u ∈ {1, 2, . . , B − 1} and v ∈ Z+ such that A · u = B · v + 1.

43n−4 , 42 · 43n−4 , n ≥ 5. From the construction above it follows that equation (1) has inﬁnitely many families of solutions with distinct components. (6) It is not known whether there are inﬁnitely many positive integers n for which equation (1) admits solutions (x1 , x2 , . . , xn ), where x1 , x2 , . . , xn are all distinct odd positive integers. A simple parity argument shows that in this case n must be odd. There are several known examples of such integers n. For instance, if n = 9, we have 1 1 1 1 1 1 1 1 1 + + + + + + + + = 1; 3 5 7 9 11 15 33 45 385 42 Part I.

Prove that there are inﬁnitely many quadruples (x, y, z, w) of positive integers such that x4 + y 4 + z 4 = 2002w . (Titu Andreescu) 10. Prove that each of the following equations has inﬁnitely many solutions in integers x, y, z, u: x2 + y 2 + z 2 = 2u2 , x4 + y 4 + z 4 = 2u2 . 11. Prove that there are inﬁnitely many quadruples (x, y, u, v) of positive integers such that xy + 1, xu + 1, xv + 1, yu + 1, yv + 1, uv + 1 are all perfect squares. 4 The Modular Arithmetic Method In many situations, simple modular arithmetic considerations are employed in proving that certain Diophantine equations are not solvable or in reducing the range of their possible solutions.