104 vấn đề lí thuyết số

[...]... property); (c) If x | y and y = 0, then |x| ≤ |y|; (d) If x | y and x | z, then x | αy + βz for any integers α and β; (e) If x | y and x | y ± z, then x | z; (f) If x | y and y | x, then |x| = |y|; 2 104 Number Theory Problems (g) If x | y and y = 0, then y x | y; (h) for z = 0, x | y if and only if x z | yz The proofs of the above properties are rather straightforward from the definition We present... the reciprocals of k odd integers? Solution: The answer is negative We approach indirectly Assume that 1= 1 1 + ··· + n1 nk for some odd integers n 1 , , n k ; then clearing denominators we obtain 4 104 Number Theory Problems n 1 · · · n k = s1 + · · · + sk , where si are all odd But this is impossible since the left-hand side is odd and the right-hand side is even If k is odd, such representations... Greek mathematician Eratosthenes (250 BCE) Note that all positive even numbers greater than 2 are composite In other words, 2 is the only even (and the smallest) prime All other primes are odd; that 6 104 Number Theory Problems is, they are not divisible by 2 The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 How many primes are there? Are we really sure that there are infinitely many primes?... + r1 )( p1 c2 + r2 ) · · · ( p1 ch + rh ) Expanding the last product we obtain n = mp1 + r1r2 · · · rh for some positive integer m Setting n = r1r2 · · · rh we have n = p1 p2 · · · pk = mp1 + n It 8 104 Number Theory Problems follows that p1 | n and n = p1 s As we have shown, s can be written as a product of primes We write s = s1 s2 · · · si , where s1 , s2 , , si are primes On the other hand,... 1)(32 − 1) = (32 + 1)(32 + 1)(32 − 1) 9 8 7 1 0 = · · · = (32 + 1)(32 + 1)(32 + 1) · · · (32 + 1)(32 + 1)(3 − 1) k By Example 1.5, 2 32 + 1, for positive integers k 9 + 2 + 1 = 12 Thus the answer is 10 104 Number Theory Problems Theorem 1.4 indicates that all integers are generated (productively) by primes Because of the importance of primes, many people have tried to find (explicit) formulas to generate... properties to the reader Proposition 1.6 (Continuation) (g) gcd(gcd(m, n), p) = gcd(m, gcd(n, p)); proving that gcd(m, n, p) is welldefined; (h) If d | ai , i = 1, , s, then d | gcd(a1 , , as ); 12 104 Number Theory Problems α α (i) If ai = p1 1i · · · pk ki , i = 1, , s, then min(α11 , ,α1k ) gcd(a1 , , as ) = p1 min(αk1 , ,αkk ) · · · pk We say that a1 , a2 , , an are relatively prime... out 2 liters of milk? Solution: Let T, L 5 , and L 9 denote the milk tank, the 5-liter container, and the 9-liter container, respectively We can use the following table to achieve the desired result 14 104 Number Theory Problems T x x −5 x −5 x − 10 x − 10 x −1 x −1 x −6 x −6 x − 11 x − 11 L5 0 5 0 5 1 1 0 5 0 5 2 L9 0 0 5 5 9 0 1 1 6 6 9 The key is to make the connection between 2 = 4 × 5 − 2 × 9 We... > g 2 x y = ab √ It follows that |a − b| > 3 ab Note that the key step x 2 + x y + y 2 divides g can also be obtained by clever algebraic manipulations such as a 3 = (a 2 + ab + b2 )a − ab(a + b) 16 104 Number Theory Problems L.C.M For a positive integer k we denote by Mk the set of all multiples of k As opposed to the set Dk defined earlier in this section, Mk is an infinite set For positive integers... answer is 4204·562 · 4202 = 4202250 Putting the last three examples together gives two interesting results in number theory For a positive integer n denote by τ (n) the number of its divisors It is 18 104 Number Theory Problems clear that τ (n) = 1 d|n Writing τ in this summation form allows us later to discuss it as an example of a multiplicative arithmetic function a a a Proposition 1.13 If n = p1... modulo m and we write a ≡ b (mod m) Proposition 1.18 (a) a ≡ a (mod m) (reflexivity) (b) If a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m) (transitivity) (c) If a ≡ b (mod m), then b ≡ a (mod m) 20 104 Number Theory Problems (d) If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m) and a − c ≡ b − d (mod m) (e) If a ≡ b (mod m), then for any integer k, ka ≡ kb (mod m) (f) If a ≡ b (mod m) . Mathe- matics Teaching from the MAA in 1996 and 2002. Titu Andreescu Dorin Andrica Zuming Feng 104 Number Theory Problems From the Training of the USA IMO Team Birkh ¨ auser Boston • Basel • Berlin Titu. as to whether or not they are subject to proprietary rights. 987654321 www.birkhauser.com (EB) 104 Number Theory Problems Titu Andreescu, Dorin Andrica, Zuming Feng October 25, 2006 Contents Preface. to Advanced Problems 131 Glossary 189 Further Reading 197 Index 203 Preface This book contains 104 of the best problems used in the training and testing of the U.S. International Mathematical