# k, then n = p2α 1 1 pα k

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square if and only if all exponents in its prime factorization are even. An integer n is a perfect power if n = m s for some integers m and s, s. Similarly, n is an sth perfect power if and only if all exponents in its prime factorization are divisible by s. We say that the integer n is square-free if for any prime divisor p, p does not divide n. Similarly, we can define the sth power-free integers. These preliminary considerations seem trivial, but as you will see shortly, they have significant rich applications in solving various problems..1 Perfect Squares Problem.1.1. Find all nonnegative integers n such that there are integers a and b with the property n = a + b and n 3 = a + b. (004 Romanian Mathematical Olympiad) Solution. From the inequality (a + b ) (a + b) we get n 3 n 4, that is, n. Thus: for n = 0, we choose a = b = 0, for n = 1, we take a = 1, b = 0, and for n =, we may take a = b =. Problem.1.. Find all integers n such that n 50 and n + 50 are both perfect squares. T. Andreescu and D. Andrica, Number Theory, DOI: /b11856_, Birkhäuser Boston, a part of Springer Science + Business Media, LLC

2 48 I Fundamentals,. Powers of Integers Solution. Let n 50 = a and n + 50 = b. Then b a = 100, so (b a)(b + a) = 5. Because b a and b+a are of the same parity, we have the following possibilities: b a =, b + a = 50, yielding b = 6, a = 4, and b a = 10, b+a = 10 with a = 0, b = 10. Hence the integers with this property are n = 66 and n = 50. Problem.1.3. Let n 3 be a positive integer. Show that it is possible to eliminate at most two numbers among the elements of the set {1,,...,n such that the sum of the remaining numbers is a perfect square. (003 Romanian Mathematical Olympiad) Solution. Let m = n(n + 1)/. From m n(n + 1)/ <(m + 1) we obtain n(n + 1) m <(m + 1) m = m + 1. Therefore, we have n(n + 1) m m n + n n 1. Since any number k, k n 1, can be obtained by adding at most two numbers from {1,,...,n, we obtain the result. Problem.1.4. Let k be a positive integer and a = 3k + 3k + 1. (i) Show that a and a are sums of three perfect squares. (ii) Show that if a is a divisor of a positive integer b, and b is a sum of three perfect squares, then any power b n is a sum of three perfect squares. (003 Romanian Mathematical Olympiad) Solution. (i) a = 6k +6k + = (k +1) +(k +1) +k and a = 9k 4 +18k k +6k +1 = (k +k) +(k +3k +1) +k (k +1) = a1 +a +a 3. (ii) Let b = ca. Then b = b1 +b +b 3 and b = c a = c (a1 +a +a 3 ). To end the proof, we proceed as follows: for n = p + 1wehaveb p+1 = (b p ) (b1 + b + b 3 ), and for n = p +, b n = (b p ) b = (b p ) c (a1 + a + a 3 ). Problem.1.5. (a) Let k be an integer number. Prove that the number (k + 1) 3 (k 1) 3 is the sum of three squares. (b) Let n be a positive number. Prove that the number (n + 1) 3 can be represented as the sum of 3n 1 squares greater than 1. (000 Romanian Mathematical Olympiad)

3 .1. Perfect Squares 49 Solution. (a) It is easy to check that (b) Observe that (k + 1) 3 (k 1) 3 = (4k) + (k + 1) + (k 1). (n + 1) 3 1 = (n + 1) 3 (n 1) 3 + (n 1) 3 (n 3) Each of the n differences in the right-hand side can be written as a sum of three squares greater than 1, except for the last one: It follows that (n + 1) 3 = = n [ (4k) + (k + 1) + (k 1) ] k= as desired. Problem.1.6. Prove that for any positive integer n the number ( ) n ( ) n is an integer but not a perfect square. Solution. Note that = + 1 ) 4 and 17 1 = 1 ) 4,so ( ) n ( ) n ) 4n ) 4n = 4 ) n ) n ) n ) n =. Define ) n ) n A = ) n ) n and B =. Using the binomial expansion formula we obtain positive integers x and y such that + 1 ) n = x + y, 1 ) n = x y. Then ) n ) n x = = A

4 50 I Fundamentals,. Powers of Integers and ) n ) n y = and so AB is as integer, as claimed. Observe that = B, A B = (A + B)(A B) = ( + 1) n ( 1) n = 1, so A and B are relatively prime. It is sufficient to prove that at least one of them is not a perfect square. We have A = + 1 ) n + 1 ) n = [ + 1 ) n + 1 ) n ] 1 (1) and A = + 1 ) n + 1 ) n = [ + 1 ) n 1 ) n ] + 1. () Since one of the numbers + 1 ) n + 1 ) n, + 1 ) n 1 ) n is an integer, depending on the parity of n, from the relations (1) and () we derive that A is not a square. This completes the proof. Problem.1.7. The integers a and b have the property that for every nonnegative integer n, the number n a + b is a perfect square. Show that a = 0. (001 Polish Mathematical Olympiad) Solution. If a = 0 and b = 0, then at least one of 1 a + b and a + b is not a perfect square, a contradiction. If a = 0andb = 0, then each (x n, y n ) = ( n a + b, n+ a + b) satisfies (x n + y n )(x n y n ) = 3b. Hence, x + n + y n divides 3b for each n. But this is impossible because 3b = 0 but x n + y n > 3b for large enough n. Therefore, a = 0. Remark. We invite the courageous reader to prove that if f Z[X] is a polynomial and f ( n ) is a perfect square for all n, then there is g Z[X] such that f = g.

5 .1. Perfect Squares 51 Problem.1.8. Prove that the number is a perfect square. First solution {{... {{ N = {{ {{ Second solution. Note that = 1 9 ( ) ( ) = 1 9 ( ) = [ 1 3 ( ) ] 1997 { { = = {{ N = {{ {{ 5 = = ( ) ; hence N is a square. Problem.1.9. Find all positive integers n, n 1, such that n + 3 n is a perfect square. Solution. Let m be a positive integer such that m = n + 3 n. Since (m n)(m+n) = 3 n, there is k 0 such that m n = 3 k and m+n = 3 n k. From m n < m + n follows k < n k, and so n k 1. If n k = 1, then n = (m + n) (m n) = 3 n k 3 k = 3 k (3 n k 1) = 3 k (3 1 1) = 3 k,so n = 3 k = k + 1. We have 3 m = (1 + ) m = 1 + m + ( m ) + > m + 1. Therefore k = 0ork = 1, and consequently n = 1orn = 3. If n k > 1, then n k and k n k. It follows that 3 k 3 n k, and consequently n = 3 n k 3 k 3 n k 3 n k = 3 n k (3 1) = 8 3 n k 8[1 + (n k )] =16n 16k 4, which implies 8k +1 7n. On the other hand, n k +; hence 7n 14k +14, contradiction. In conclusion, the only possible values for n are 1 and 3.

6 5 I Fundamentals,. Powers of Integers Problem Find the number of five-digit perfect squares having the last two digits equal. Solution. Suppose n = abcdd is a perfect square. Then n = 100abc + 11d = 4m + 3d for some m. Since all squares have the form 4m or 4m + 1 and d {0, 1, 4, 5, 6, 9 as the last digit of a square, it follows that d = 0ord = 4. If d = 0, then n = 100abc is a square if abc is a square. Hence abc {10,11,...,31, so there are numbers. If d = 4, then 100abc+44 = n = k implies k = p and abc = p (1) If p = 5x, then abc is not an integer, false. () If p = 5x + 1, then abc = 5x +10x 1 5 = x + (x 1) 5 x {11, 16, 1, 6, 31, so there are 5 solutions. (3) If p = 5x +, then abc = x + 0x 7 5 N, false. (4) If p = 5x +3, then abc = x + 30x 5 N, false. (5) If p = 5x +4 then abc = x + 8x+1 5 ; hence x = 5m + 3 for some m x {13, 18, 3, 8, so there are four solutions. Finally, there are = 31 squares. Problem The last four digits of a perfect square are equal. Prove they are all zero. (00 Romanian Team Selection Test for JBMO) Solution. Denote by k the perfect square and by a the digit that appears in the last four positions. It easily follows that a is one of the numbers 0, 1, 4, 5, 6, 9. Thus k a 1111 (mod 16). (1) If a = 0, we are done. () Suppose that a {1, 5, 9. Since k 0 (mod 8), k 1 (mod 8) or k 4 (mod 8) and (mod 8), we obtain (mod 8), (mod 8), and (mod 8). Thus the congruence k a 1111 (mod 16) cannot hold. (3) Suppose a {4, 6. Since (mod 16), (mod 16), and (mod 16), we conclude that in this case the congruence k a 1111 (mod 16) cannot hold. Thus a = 0. Remark. 38 = 1444 ends in three equal digits, so the problem is sharp. Problem.1.1. Let 1 < n 1 < n < < n k < be a sequence of integers such that no two are consecutive. Prove that for all positive integers m between n 1 + n + +n m and n + n + +n m+1 there is a perfect square. Solution. It is easy to prove that between numbers a > b 0 such that a b > 1 there is a perfect square: take for example ([ b]+1). It suffices to prove that n1 + +n m+1 n 1 + +n m > 1, m 1. This is equivalent to n 1 + +n m + n m+1 >(1 + n 1 + n + +n m ), and then n m+1 > 1 + n 1 + n + +n m, m 1.

7 .1. Perfect Squares 53 We induct on m. Form = 1 we have to prove that n > 1 + n 1. Indeed, n > n 1 + = 1 + (1 + n 1 )>1 + n 1. Assume that the claim holds for some m 1. Then n m+1 1 > n 1 + +n m so (n m+1 1) > 4(n 1 + +n m ) hence This implies (n m+1 + 1) > 4(n 1 + +n m+1 ). n m > n 1 + +n m+1, and since n m+ n m+1, it follows that n m+ > 1 + n 1 + +n m+1, as desired. Problem Find all integers x, y, z such that 4 x + 4 y + 4 z is a square. Solution. It is clear that there are no solutions with x < 0. Without loss of generality assume that x y z and let 4 x +4 y +4 z = u. Then x (1+4 y x +4 z x ) = u. We have two situations. Case y x + 4 z x is odd, i.e., y x + 4 z x = (a + 1). It follows that 4 y x z x 1 = a(a + 1), and then 4 y x 1 (1 + 4 z y ) = a(a + 1). We consider two cases. (1) The number a is even. Then a +1 is odd, so 4 y x 1 = a and z y = a + 1. It follows that 4 y x 1 = 4 z y ; hence y x 1 = z y. Thus z = y x 1 and 4 x + 4 y + 4 z = 4 x + 4 y + 4 y x 1 = ( x + y x 1 ). () The number a is odd. Then a + 1 is even, so a = 4 z y + 1, a + 1 = 4 y x 1 and 4 y x 1 4 z y =. It follows that y x 3 = x y 1 + 1, which is impossible, since x y 1 = 0. Case y x + 4 z x is even; thus y = x or z = x. Anyway, we must have y = x, and then + 4 z x is a square, which is impossible, since it is congruent to (mod 4) or congruent to 3 (mod 4).

8 54 I Fundamentals,. Powers of Integers Additional Problems Problem Let x, y, z be positive integers such that 1 x 1 y = 1 z. Let h be the greatest common divisor of x, y, z. Prove that hxyz and h(y x) are perfect squares. (1998 United Kingdom Mathematical Olympiad) Problem Let b an integer greater than 5. For each positive integer n, consider the number x n = 11 {{...1 {{... 5, n 1 n written in base b. Prove that the following condition holds if and only if b = 10: There exists a positive integer M such that for every integer n greater than M, the number x n is a perfect square. (44th International Mathematical Olympiad Shortlist) Problem Do there exist three natural numbers greater than 1 such that the square of each, minus one, is divisible by each of the others? (1996 Russian Mathematical Olympiad) Problem (a) Find the first positive integer whose square ends in three 4 s. (b) Find all positive integers whose squares end in three 4 s. (c) Show that no perfect square ends with four 4 s. (1995 United Kingdom Mathematical Olympiad) Problem Let abc be a prime. Prove that b 4ac cannot be a perfect square. (Mathematical Reflections) Problem For each positive integer n, denote by s(n) the greatest integer such that for all positive integer k s(n), n can be expressed as a sum of squares of k positive integers. (a) Prove that s(n) n 14 for all n 4. (b) Find a number n such that s(n) = n 14. (c) Prove that there exist infinitely many positive integers n such that s(n) = n 14. (33rd International Mathematical Olympiad)

9 .1. Perfect Squares 55 Problem.1.0. Let A be the set of positive integers representable in the form a + b for integers a, b with b = 0. Show that if p A for a prime p, then p A. (1997 Romanian International Mathematical Olympiad Team Selection Test) Problem.1.1. Is it possible to find 100 positive integers not exceeding 5000 such that all pairwise sums of them are different? (4nd International Mathematical Olympiad Shortlist) Problem.1.. Do there exist 10 distinct integers, the sum of any 9 of which is a perfect square? (1999 Russian Mathematical Olympiad) Problem.1.3. Let n be a positive integer such that n is a divisor of the sum Prove that n is square-free. n i n 1. i=1 (1995 Indian Mathematical Olympiad) Problem.1.4. Let n, p be integers such that n > 1andp is a prime. If n (p 1) and p (n 3 1), show that 4p 3 is a perfect square. (00 Czech Polish Slovak Mathematical Competition) Problem.1.5. Show that for any positive integer n > 10000, there exists a positive integer m that is a sum of two squares and such that 0 < m n < 3 4 n. (Russian Mathematical Olympiad) Problem.1.6. Show that a positive integer m is a perfect square if and only if for each positive integer n, at least one of the differences is divisible by n. (m + 1) m, (m + ) m,..., (m + n) m (00 Czech and Slovak Mathematical Olympiad)

10 56 I Fundamentals,. Powers of Integers. Perfect Cubes Problem..1. Prove that if n is a perfect cube, then n + 3n + 3 cannot be a perfect cube. Solution. If n = 0, then we get 3 and the property is true. Suppose by way of contradiction that n + 3n + 3 is a cube for some n = 0. Hence n(n + 3n + 3) is a cube. Note that n(n + 3n + 3) = n 3 + 3n + 3n = (n + 1) 3 1, and since (n + 1) 3 1 is not a cube when n = 0, we obtain a contradiction. Problem... Let m be a given positive integer. Find a positive integer n such that m + n + 1 is a perfect square and mn + 1 is a perfect cube. Solution. Choosing n = m + 3m + 3, we have and m + n + 1 = m + 4m + 4 = (m + ) mn + 1 = m 3 + 3m + 3m + 1 = (m + 1) 3. Problem..3. Which are there more of among the natural numbers from 1 to , inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be? (1996 Russian Mathematical Olympiad) Solution. There are more numbers not of this form. Let n = k + m 3, where k, m, n N and n Clearly k 1000 and m 100. Therefore there cannot be more numbers in the desired form than the pairs (k, m). Problem..4. Show that no integer of the form xyxy in base 10 can be the cube of an integer. Also find the smallest base b > 1 in which there is a perfect cube of the form xyxy. (1998 Irish Mathematical Olympiad) Solution. If the 4-digit number xyxy = 101 xy is a cube, then 101 xy, which is a contradiction. Convert xyxy = 101 xy from base b to base 10. We obtain xyxy = (b +1) (bx + y) with x, y < b and b +1 > bx + y. Thus for xyxy to be a cube, b + 1 must be divisible by a perfect square. We can check easily that b = 7 is the smallest such number, with b + 1 = 50. The smallest cube divisible by 50 is 1000, which is 66 is base 7.

11 Additional Problems.3. kth Powers of Integers, k at least 4 57 Problem..5. Find all the positive perfect cubes that are not divisible by 10 such that the number obtained by erasing the last three digits is also a perfect cube. Problem..6. Find all positive integers n less than 1999 such that n is equal to the cube of the sum of n s digits. (1999 Iberoamerican Mathematical Olympiad) Problem..7. Prove that for any nonnegative integer n the number A = n + 3 n + 5 n + 6 n is not a perfect cube. Problem..8. Prove that every integer is a sum of five cubes. Problem..9. Show that every rational number can be written as a sum of three cubes..3 kth Powers of Integers, k at least 4 Problem.3.1. Given 81 natural numbers whose prime divisors belong to the set {, 3, 5, prove that there exist four numbers whose product is the fourth power of an integer. (1996 Greek Mathematical Olympiad) Solution. It suffices to take 5 such numbers. To each number, associate the triple (x, x 3, x 5 ) recording the parity of the exponents of, 3, and 5 in its prime factorization. Two numbers have the same triple if and only if their product is a perfect square. As long as there are 9 numbers left, we can select two whose product is a square; in so doing, we obtain 9 such pairs. Repeating the process with the square roots of the products of the pairs, we obtain four numbers whose product is a fourth power. Problem.3.. Find all collections of 100 positive integers such that the sum of the fourth powers of every four of the integers is divisible by the product of the four numbers. (1997 St. Petersburg City Mathematical Olympiad) Solution. Such sets must be n, n,...,n or 3n, n, n,...,n for some integer n. Without loss of generality, we assume that the numbers do not have a common factor. If u,v,w,x, y are five of the numbers, then uvw divides u 4 +v 4 +w 4 +x 4 and u 4 + v 4 + w 4 + y 4, and so divides x 4 y 4. Likewise, v 4 w 4 x 4 (mod u), and from above, 3v 4 0 (mod u). Ifu has a prime divisor not equal

12 58 I Fundamentals,. Powers of Integers to 3, we conclude that every other integer is divisible by the same prime, contrary to assumption. Likewise, if u is divisible by 9, then every other integer is divisible by 3. Thus all of the numbers equal 1 or 3. Moreover, if one number is 3, the others are all congruent modulo 3, so are all 3 (contrary to assumption) or 1. This completes the proof. Problem.3.3. Let M be a set of 1985 distinct positive integers, none of which has a prime divisor greater than 6. Prove that M contains at least one subset of four distinct elements whose product is the fourth power of an integer. (6th International Mathematical Olympiad) Solution. There are nine prime numbers less than 6: p 1 =, p = 3,..., p 9 = 3. Any element x of M has a representation x = 9 i=1 p a i i, a i 0. If x, y M and y = 9 i=1 p b i i, the product xy = 9 i=1 p a i +b i i is a perfect square if and only if a i + b i 0 (mod ). Equivalently, a i b i (mod ) for all i = 1,,...,9. Because there are 9 = 51 elements in (Z/Z) 9, any subset of M having at least 513 elements contains two elements x, y such that xy is a perfect square. Starting from M and eliminating such pairs, one obtains 1 ( ) = 736 > 513 distinct two-element subsets of M having a square as the product of elements. Reasoning as above, we find among these squares at least one pair (in fact many pairs) whose product is a fourth power. Problem.3.4. Let A be a subset of {0, 1,...,1997 containing more than 1000 elements. Prove that A contains either a power of, or two distinct integers whose sum is a power of. (1997 Irish Mathematical Olympiad) Solution. Suppose A did not satisfy the conclusion. Then A would contain at most half of the integers from 51 to 1997, since they can be divided into pairs whose sum is 048 (with 104 left over); likewise, A contains at most half of the integers from 14 to 50, at most half of the integers from 3 to 13, and possibly 0, for a total of = 997 integers. Problem.3.5. Show that in the arithmetic progression with first term 1 and difference 79, there are infinitely many powers of 10. (1996 Russian Mathematical Olympiad) First solution. We will show that for all natural numbers n,10 81n 1 is divisible by 79. In fact, 10 81n 1 = (10 81 ) n 1 n = ( ) A,

13 .3. kth Powers of Integers, k at least 4 59 and = {{ 81 = (9...9 {{ ) ( {{ {{ {{ ) = 9(1...1 {{ ) ( {{ {{ {{ ) The second and third factors have nine digits equal to 1 and the root of digits (if any) 0, so the sum of the digits is 9, and each is a multiple of 9. Hence is divisible by 9 3 = 79, as is 10 81n 1 for any n. Second solution. In order to prove that is divisible by 9 3, just write ( ) ( ) = (9 + 1) 81 1 = k = k = (k + 361) 9 3. Remark. An alternative solution uses Euler s theorem (see Section 7.). We have 10 ϕ(79) 1 (mod 79); thus 10 nϕ(79) is in this progression for any positive integer n. Additional Problems Problem.3.6. Let p be a prime number and a, n positive integers. Prove that if then n = 1. p + 3 p = a n, Problem.3.7. Let x, y, p, n, k be natural numbers such that x n + y n = p k. (1996 Irish Mathematical Olympiad) Prove that if n > 1 is odd and p is an odd prime, then n is a power of p. (1996 Russian Mathematical Olympiad) Problem.3.8. Prove that a product of three consecutive integers cannot be a power of an integer. Problem.3.9. Show that there exists an infinite set A of positive integers such that for any finite nonempty subset B A, x B x is not a perfect power. (Kvant) Problem Prove that there is no infinite arithmetic progression consisting only of powers.

14

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Some practice problems for midterm 2

Some practice problems for midterm 2 Kiumars Kaveh November 15, 2011 Problem: What is the remainder of 6 2000 when divided by 11? Solution: This is a long-winded way of asking for the value of 6 2000 mod

### SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me

SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### 1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### Problem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB

Problem Set 5 1. (a) Four-digit number S = aabb is a square. Find it; (hint: 11 is a factor of S) (b) If n is a sum of two square, so is 2n. (Frank) Solution: (a) Since (A + B) (A + B) = 0, and 11 0, 11

### b) Find smallest a > 0 such that 2 a 1 (mod 341). Solution: a) Use succesive squarings. We have 85 =

Problem 1. Prove that a b (mod c) if and only if a and b give the same remainders upon division by c. Solution: Let r a, r b be the remainders of a, b upon division by c respectively. Thus a r a (mod c)

### On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

### PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

### Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

### Introduction. Appendix D Mathematical Induction D1

Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Pythagorean Triples Pythagorean triple similar primitive

Pythagorean Triples One of the most far-reaching problems to appear in Diophantus Arithmetica was his Problem II-8: To divide a given square into two squares. Namely, find integers x, y, z, so that x 2

### PEN A Problems. a 2 +b 2 ab+1. 0 < a 2 +b 2 abc c, x 2 +y 2 +1 xy

Divisibility Theory 1 Show that if x,y,z are positive integers, then (xy + 1)(yz + 1)(zx + 1) is a perfect square if and only if xy +1, yz +1, zx+1 are all perfect squares. 2 Find infinitely many triples

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

### MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### Sets and Counting. Let A and B be two sets. It is easy to see (from the diagram above) that A B = A + B A B

Sets and Counting Let us remind that the integer part of a number is the greatest integer that is less or equal to. It is denoted by []. Eample [3.1] = 3, [.76] = but [ 3.1] = 4 and [.76] = 6 A B Let A

### The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

### Settling a Question about Pythagorean Triples

Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-Mail address:

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### Today s Topics. Primes & Greatest Common Divisors

Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

### Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### The Locker Problem. Bruce Torrence and Stan Wagon

The Locker Problem Bruce Torrence and Stan Wagon The locker problem appears frequently in both the secondary and university curriculum [2, 3]. In November 2005, it appeared on National Public Radio s Car

### 10 k + pm pm. 10 n p q = 2n 5 n p 2 a 5 b q = p

Week 7 Summary Lecture 13 Suppose that p and q are integers with gcd(p, q) = 1 (so that the fraction p/q is in its lowest terms) and 0 < p < q (so that 0 < p/q < 1), and suppose that q is not divisible

### Congruent Numbers, the Rank of Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture. Brad Groff

Congruent Numbers, the Rank of Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture Brad Groff Contents 1 Congruent Numbers... 1.1 Basic Facts............................... and Elliptic Curves.1

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Winter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com

Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).

### MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public

### Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

### PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

### 2 When is a 2-Digit Number the Sum of the Squares of its Digits?

When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Appendix F: Mathematical Induction

Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another

### On the largest prime factor of x 2 1

On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Consequently, for the remainder of this discussion we will assume that a is a quadratic residue mod p.

Computing square roots mod p We now have very effective ways to determine whether the quadratic congruence x a (mod p), p an odd prime, is solvable. What we need to complete this discussion is an effective

### MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.

MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.

### Congruences. Robert Friedman

Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### Collinear Points in Permutations

Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,

### Solutions to Problems for Mathematics 2.5 DV Date: April 14,

Solutions to Problems for Mathematics 25 DV Date: April 14, 2010 1 Solution to Problem 1 The equation is 75 = 12 6 + 3 Solution to Problem 2 The remainder is 1 This is because 11 evenly divides the first

### CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Integers and division

CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.

### This section demonstrates some different techniques of proving some general statements.

Section 4. Number Theory 4.. Introduction This section demonstrates some different techniques of proving some general statements. Examples: Prove that the sum of any two odd numbers is even. Firstly you

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Chapter Two. Number Theory

Chapter Two Number Theory 2.1 INTRODUCTION Number theory is that area of mathematics dealing with the properties of the integers under the ordinary operations of addition, subtraction, multiplication and

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Category 3 Number Theory Meet #1, October, 2000

Category 3 Meet #1, October, 2000 1. For how many positive integral values of n will 168 n be a whole number? 2. What is the greatest integer that will always divide the product of four consecutive integers?

### NUMBER THEORY AMIN WITNO

NUMBER THEORY AMIN WITNO ii Number Theory Amin Witno Department of Basic Sciences Philadelphia University JORDAN 19392 Originally written for Math 313 students at Philadelphia University in Jordan, this

### ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM

Acta Math. Univ. Comenianae Vol. LXXXI, (01), pp. 03 09 03 ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM A. DUBICKAS and A. NOVIKAS Abstract. Let E(4) be the set of positive integers expressible

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### PRIME FACTORS OF CONSECUTIVE INTEGERS

PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in

### Characterizing the Sum of Two Cubes

1 3 47 6 3 11 Journal of Integer Sequences, Vol. 6 (003), Article 03.4.6 Characterizing the Sum of Two Cubes Kevin A. Broughan University of Waikato Hamilton 001 New Zealand kab@waikato.ac.nz Abstract

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### ( ) FACTORING. x In this polynomial the only variable in common to all is x.

FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

### Lectures on Number Theory. Lars-Åke Lindahl

Lectures on Number Theory Lars-Åke Lindahl 2002 Contents 1 Divisibility 1 2 Prime Numbers 7 3 The Linear Diophantine Equation ax+by=c 12 4 Congruences 15 5 Linear Congruences 19 6 The Chinese Remainder

### The Chinese Remainder Theorem

The Chinese Remainder Theorem Evan Chen evanchen@mit.edu February 3, 2015 The Chinese Remainder Theorem is a theorem only in that it is useful and requires proof. When you ask a capable 15-year-old why

### MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

### 1. By how much does 1 3 of 5 2 exceed 1 2 of 1 3? 2. What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3.

1 By how much does 1 3 of 5 exceed 1 of 1 3? What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3 A ticket fee was \$10, but then it was reduced The number of customers

### 12 Greatest Common Divisors. The Euclidean Algorithm

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to