Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Pythagorean Quadruples 1. Pythagorean and Primitive Pythagorean Quadruples A few years ago, I became interested with those 3 vectors with integer coordinates and integer length or norm. The simplest example is the vector 2i + 2 j + k whose length is 3. A couple of other elementary examples include 2i + 6 j + 3k of norm 7 and 3i + 4 j +12k of norm 13. Of course these integer friendly vectors generalizes Pythagorean triples: An ordered 4-tuple of positive integers ( a,b,c,d ) is a Pythagorean Quadruple (PQ) if a 2 + b 2 + c 2 = d 2. This quadruple is primitive (PPQ) provided that gcd(a,b,c,d) =1. It is easy to see that a PQ is a multiple of PPQ and a PQ (a,b,c,d) is a PPQ if the gcd of any three of these positive integers is 1, in particular if gcd(a,b,c) = 1. In this article I would like to share my discovery of a formula that generates all PPQ s. It should be noted that in the 18th and 19th centuries several formulas were discovered to generate PQ s; however, most of them did not generate all PQ s or PPQ s. The first step I took to discover a formula for PPQ s was to examine Euclid's Theorem on Primitive Pythagorean Triples: The Primitive Pythagorean Triple Theorem Let a, b and c be positive integers such that a 2 + b 2 = c 2. Also assume that gcd(a,b,c) = 1. Then c is odd and either a or b is odd (but not both; say, b is odd and a is even). Then there exist relatively prime positive integers u and v of opposite parity, with u > v, such that:, and. Furthermore, all such triples (a,b,c) described above are Primitive Pythagorean Triples as well.

2 This theorem deserves a few observations: 1. There is a parity condition on the generators u and v. (Another way to express "u and v have opposite parity" is to write "u+v is odd.") 2. The Pythagorean triple itself enjoys a couple of parity conditions: c is odd and a+b is also odd. 3. We also have an algebraic identity: ; i.e., this equation is true regardless of any conditions placed on u and v. The first step I took in my search for a formula was to see if PPQ s had any parity conditions. To this end, I examined a few PPQ's : (1,2,2,3), (2,3,6,7), (3,4,12,13). It appears that d is always odd, one of a, b or c is also odd and the other two are even. This observation turns out to be true and is easily proved using mod 4 arithmetic. The Parity Lemma for PPQ's Let be a PPQ. Then d is odd, and exactly one of a, b or c is odd as well. I will now adopt the convention whenever is a PPQ, then a and b are even and c and (of course) d are odd. The next step was to find an algebraic identity that would generate (not necessarily primitive) PQ s. The following is an "obvious" and wellknown generalization of the Pythagorean Triple identity: Let u, v, w be positive integers, u 2 +w 2 > v 2. Now suppose that and. Then is a PQ.

3 It should be clear that a necessary condition for this PQ to be primitive is that gcd(u,v,w) = 1. (We might also conjecture a parity condition, namely, u+v+w is odd.) We will now collect some data using this identity and make some observations concerning the gcd = gcd(a,b,c,d). u w v a b c d gcd 1) ) ) ) ) ) ) This table demonstrates that the two conditions, gcd(u,v,w) = 1 and u+v+w is odd, are not sufficient to guarantee that (u,v,w) generates a PPQ. Indeed, there do not exist integer generators for the primitive parents of the PQ's listed in (3), (4) and (5). (The primitive parents of (3), (4) and (5) are,

4 respectively, (4,28,35,45), (20,160,73,177) and (24,68,45,95)). It appears that we need to include a divisor along with our generators to insure ourselves of a PPQ. We now come to the task of relating the gcd to its generators. At this point, it is difficult to motivate how one would discover the relationship. It suffices to say that after staring at these examples for several days and a long walk I discovered that: gcd(a,b,c,d) = gcd(u 2 + w 2, v). In our examples (3), (4) and (5) we have: gcd(20,140,175,225) = 5 = gcd(200,5) = gcd( , 5) gcd(260,2080,949,2301) =13 = gcd(1625,26) = gcd( ,26) gcd(120,240,225,425) = 5 = gcd(325,10) = gcd( ,10). Note that we do not have gcd(a,b,c,d) = gcd(u 2 + w 2, v 2 ), as illustrated in example (5): gcd( ,10 2 ) = gcd(325,100) =25. Finally, we need to find a parity condition for the generators. To do this, we will examine examples (6) and (7). Both of these PQ's, (12,4,6,14) and (6,4,12,14), are essentially the same. However, in example (7) we see that: gcd(u 2 + w 2, v) = gcd(13,1) = 1 2 =gcd(12,4,6,14). On the other hand, example (6) does satisfy the gcd conjecture: gcd(u 2 + w 2, v) = gcd(10,2) = 2 = gcd(12,4,6,14). Furthermore, the primitive parent of example (6), (6,2,3,7), satisfies our parity convention for PPQ's (a and b are even, c and d are odd). The primitive parent of example (7), (3,2,6,7), does not satisfy our parity convention. The important feature to notice here, in example (7), is that: v = 1 is odd and u+w = 5 is also odd (i.e., not even). Looking at other examples where v is odd we see that in examples (1), (2) and (3) that u+w is even. In the examples where v is even, it does not matter whether u+w is even or odd;

5 each of these examples does satisfy our gcd conjecture and their primitive parents satisfy our parity convention. We can now state our parity condition for the generators (u,v,w) as follows: If v is odd, then u+w is even. 2. The Primitive Pythagorean Quadruple Theorems We are now in a position to state and prove the Primitive Pythagorean Quadruple Theorem. Actually, we will accomplish this via two theorems: (I) The Verification Theorem. All generators (u,v,w), gcd(u,v,w) =1, satisfying the parity condition together with the divisor g will provide us with a Primitive Pythagorean Quadruple. (II) The Generator Theorem. Every PPQ has generators (u,v,w), gcd(u,v,w) = 1, satisfying the parity condition and a divisor g. As you are about to see, the proof of the Verification Theorem is not nearly as simple and straightforward as it is in the verification for Primitive Pythagorean Triples. The Verification Theorem Let u, v, w be positive integers,, and let g = gcd(u 2 + w 2, v). Suppose that: (i) gcd(u,v,w) = 1. (ii) If v is odd, then u+w is even. (iii) a = 2uv/g, b = 2wv/g, c = (u 2 + w 2 - v 2 )/g and d = (u 2 + w 2 + v 2 )/g. Then (a,b,c,d) is a PPQ with a and b even and c and d odd. Proof. A simple computation shows that (a,b,c,d) is a PQ. We will now show that a and b are even. Since g divides v, (v/g) is a positive integer. Therefore, a = 2u(v/g) and b = 2w(v/g) are even. Next, we demonstrate that d is odd. There are three cases to consider.

6 case(i) Suppose that v is odd. Then u 2 + w 2 is even since u+w is even. Hence, dg = u 2 + w 2 + v 2 is odd. This implies that d is odd. case(ii) Suppose that v is even and u + w is also even. Then u and w are both odd since gcd(u,v,w) = 1. Thus, dg = u 2 + w 2 + v 2 2(mod 4). Note that g is even since v and u 2 + w 2 are both even. If d were even, then dg would be a multiple of 4, contradicting the above congruence. Hence, d must be odd. case(iii) Suppose that v is even and u+w is odd. Then u 2 + w 2 is odd. Therefore, dg = u 2 + w 2 + v 2 1(mod 2) and d must be odd in this case as well. It now follows that c is odd since is odd. Our final task is to prove that our PQ is primitive. As is typical in these situations, we shall assume otherwise and then derive some sort of contradiction. If (a,b,c,d) were not primitive, then there would be a prime p that divides each of a, b, c and d. ( Note that p is odd since p divides d and d is odd.) Clearly, p divides each of 2uv, 2wv, and since 2uv = ag, 2wv = bg, u 2 + w 2 v 2 = cg and u 2 + w 2 + v 2 = dg. Therefore p divides dg cg = 2v 2. Consequently p divides v. Similarly p divides u 2 + w 2. Hence p divides g because g = gcd(u 2 + w 2,v). Since gcd(u,v,w) = 1 and p divides v, either p does not divide u or p does not divide w; let us suppose that p does not divide u. Now p divides a and p divides g implies that p 2 divides ag, i.e., p 2 divides 2uv. Hence, p 2 divides v. Also, p divides g and p divides d implies that p 2 divides dg, i.e., p 2 divides u 2 + w 2 + v 2. We now have p 2 dividing both v and u2 + w2 + v2, thus p2 divides u2 + w2. Hence p2 divides g. Repeating this argument and using mathematical induction, we conclude that p n divides

7 v and g for all positive integers n. This is absurd. The proof of the verification theorem is now complete. We now come to the vexing task of when given a PPQ (a,b,c,d), how do we find its generators (u,v,w) and its divisor g? Working backwards, we can solve for (the squares) of u, w and v:, and. The important feature to observe here is that g times each of the expressions,, is a (perfect) square. There is an abundance of positive integers with this property, e.g., the number has this property. Our divisor g has to be large enough in order to provide us with squares and yet small enough to divide 2uv, 2wv, and. Perhaps the smallest positive integer times each of, which produces squares, is the number "g" we seek. The Generator Theorem Let (a,b,c,d) be a PPQ. Then there exists positive integers u, v and w, u2 + w 2 > v 2, and a positive integer g such that: (i) a = 2uv/g, b = 2wv/g, c = and d =. (ii) gcd(u,v,w) = 1 (iii) If v is odd, then is also odd. Proof. Let S be the set of all positive integers x such that: and are all squares. Note that S is nonempty since

8 is a member of S. By the Well Ordering Principle, S has a smallest member, say g. We now define:, and. A straightforward computation demonstrates that (i) holds. We now show that (ii) holds as well, i.e., gcd(u,v,w) = 1. Otherwise there would be a prime p such that p divides each of u, v and w (thus, u/p, v/p and w/p are all positive integers). Therefore, p 2 divides the integers ag, bg and cg since ag = 2uv, bg = 2wv and cg = u 2 + w 2 v 2. By virtue of the gcd(a,b,c) = 1, p does not divide a or p does not divide b or p does not divide c. Hence p 2 divides g, say g/p 2 = h ε Z +. Note that h < g. We now have: This is a contradiction since g was chosen as the smallest positive integer with this property. This proves (ii). To prove (iii) note that g divides since =. (Recall that d and c are both odd, thus is an integer.) We will now assume that v is odd. Hence g is odd and therefore gd and v 2 are odd as well. Since u 2 + w 2 + v 2 = gd,. Thus,, i.e., is even. This completes the proof of the Generator Theorem.

9 3. Generalizations Is our PPQ formula a generalization of the Primitive Pythagorean Triple formula? Certainly, the algebraic identity that we used to generate PQ's is a generalization of the PT identity. Indeed, if we set w = 0 we obtain: a = 2uv/g, b = 0, c = (u 2 v 2 )/g and d = (u 2 + v 2 )/g. We can now easily show in this case that: (i) gcd(u,v) = 1, (ii) g=1 and (iii) u and v have opposite parity. (i) 1 = gcd(u,v,0) = gcd(u,v). (ii) 1 = gcd(u,v) = gcd(u 2, v) = gcd(u , v) = g. (iii) There are two cases here: v is even or v is odd. If v is even, then u must be odd since gcd(u,v) = 1. On the other hand, if v is odd, then u+w = u+0 =u is even. At this point it would be tempting to generalize the PPQ formula to higher order Pythagorean n-tuples. However, the situation for (Primitive) Pythagorean Quintuples and above is a little murky and offers its own surprises. The natural generalization for the Quintuple case would be: Let x, y, z, v be positive integers and let g=gcd(x 2 +y 2 +z 2,v). Suppose that : (i) gcd(x,y,z,v) = 1 (ii) If v is odd, then x+y+z is even (iii) a=2xv/g, b=2yv/g, c=2zv/g, d=(x 2 +y 2 +z 2 v 2 )/g and e=(x 2 +y 2 +z 2 +v 2 )/g. Then (a,b,c,d,e) is a Primitive Pythagorean Quintuple. This statement is indeed true and its proof is very similar to the Verification Theorem. For example, the generator (2,1,1,1) generates the Primitive Pythagorean Quintuple (4,2,2,5,7). However, not every Primitive Pythagorean

10 Quintuple can be generated by the above formula; for example, consider (a,b,c,d,e) = (1,1,1,1,2). Part of the problem here is that in our formula, e is always odd. This difficulty should have been anticipated since every positive integer is the sum of four (or fewer) positive squares. In particular, Primitive Pythagorean Quintuples do not enjoy a parity condition that is similar to Pythagorean Triples and Quadruples. The situation here is not hopeless; there is another algebraic identity that we may employ: (xv) 2 + (yv) 2 + (zv) 2 + [(x 2 +y 2 +z 2 v 2 )/2] 2 = [(x 2 +y 2 +z 2 +v 2 )/2] 2. For example, if we use (x,y,z,v)=(1,1,1,1), then the above formula would generate the Pythagorean Quintuple (1,1,1,1,2). We can illustrate this formula with another example: (4,2,1,1) generates (4,2,1,10,11). This Pythagorean Quintuple can, essentially, be generated by our "Quintuple Conjecture" stated earlier: (2,1,5,5) generates (4,2,10,1,11). Uniqueness of the generators can be preserved if we separate e into two cases: even or odd. The proof of the following theorem, its converse and any further generalizations is left to the enjoyment of the reader. The Pythagorean Quintuple Theorem. Let (a,b,c,d,e) be a Primitive Pythagorean Quintuple. Then there exists positive integers x,y, z and v, x 2 +y 2 +z 2 > v 2, and a positive integer g such that: (I) When e is odd (i) gcd(x,y,z,v) = 1. (ii) If v is odd, then x+y+z is even. (iii) a=2xv/g, b=2yv/g, c=2zv/g, d=(x 2 +y 2 +z 2 v 2 )/g and e=(x 2 +y 2 +z 2 +v 2 )/g. (II) When e is even (i) gcd(x,y,z,v) = 1

11 (ii) x, y,z and v are all odd (iii) a=xv/g, b=yv/g, c=zv/g, d=(x 2 +y 2 +z 2 v 2 )/(2g) and e=(x 2 +y 2 +z 2 +v 2 )/(2g). It should come as no surprise that in the converse of this theorem g = gcd(x 2 +y 2 +z 2,v). 4. A Little History The problem of finding a formula that will generate Pythagorean Quadruples has been around for over 200 years. In this section I have selected a few of these formulas that can be found in Dickson's [3] History of the Theory Numbers vol. II, chapter VII [3]. 1. Matsunago (Japan, c. 1740). Take your favorite pair of positive integers a and b, and then factor a 2 + b 2 = mn, where m > n, m+n even. Now let c = (m n)/2 and d = (m + n)/2. Then (a,b,c,d) is a PQ. To illustrate, consider a = 3 and b = 6. Then, a 2 + b 2 = 45 = 9 5. We now have the PQ (3,6,2,7). It should be noted that this method, although very simple, will not generate all PPQ's. For example the PPQ (2,28,35,45) cannot be generated using Matsunago s method. 2. H.S. Monck (Great Britain(?), 1878). Monck's algorithm is particularly amusing since you can generate a PQ from a given PQ: Given a PQ (a,b,c,d), not necessarily positive, let x=a+b+d, y=a+c+d, z=b+c+d and u=a+b+c+2d. Then (x,y,z,u) is another PQ. The example provided for us in Dickson is: From (1,-2,2,3) we obtain (2,3,6,7). 3. A. Desbouves (France, 1886)

12 Mr. Desbouves proved that all PQ's (not just PPQ's) (a,b,c,d) have the form: a=2(p 2 +q 2 s 2 ), b=2[(p s) 2 q 2 + (q s) 2 ], c = (q s) 2 p 2 + 4q(p s) 2 and d = 3[(p s) 2 + q 2 ] +2s(p q) 2. Amazing as this formula is, it is not clear how this formula generalizes the Pythagorean Triple formula. 4. In 1907 A. Hurwitz gave a somewhat complicated formula for the number of solutions (a,b,c) for the equation, a 2 + b 2 + c 2 = d 2, when d is given. 5. Another Problem A problem similar to finding Pythagorean Quadruples is the following: Does there exist a triple of positive integers (x,y,z) such that x 2 +y 2,x 2 +z 2 and y 2 +z 2 are all squares? The answer is, of course, yes. In 1719 Paul Halcke found one solution to be (44,240,117). There are several formulas to generate some of these triples, the simplest being the one offered to us by the 18th century (blind from infancy) mathematician N. Saunderson: Suppose that (a,b,c) is a Pythagorean Triple. Now let x=a(4b 2 c 2 ), y=b(4a 2 c 2 ) and z=4abc. For example, (3,4,5) generates (117,44,240). As in Pythagorean Triples and Quadruples these triples enjoy a geometrical interpretation as well: A triple of positive integers (x,y,z) with the above property corresponds to a rectangular parallelepiped of width x, length y and height z which has positive integer values for all its face diagonals. For other formulas and a short history of this problem please read chapter XIX, vol. II, PP , in Dickson. 6. The Last Problem What is meant here by "The Last Problem" is the final problem to be mentioned in this article. In the February, 1994 issue of FOCUS, Keith

13 Devlin in his monthly editorial offered the following as the next "Fermat Problem" (which he called the "Integer Brick" problem): "The problem asks if it is possible to construct a rectangular brick (i.e., a rectangular parallelepiped), all of whose diagonals (both face and cross-diagonals) are integers." Altogether this problem has seven parameters: the three dimensions, the three face diagonals and the inner or cross-diagonal. Partial solutions have been found for six of the seven parameters to be integer valued. One solution, known to Euler is (104,153,672). (Here, = 185 2, = and = ) For a discussion of these solutions the reader is invited to read problem D18 in Unsolved Problems in Number Theory, 2nd edition, by Richard K. Guy [5] or the John Leech article in the American Mathematical Monthly, vol. 84 (1977), pp [6]. References 1. David M. Burton, Elementary Number Theory, 4th ed., McGraw-Hill, New York, James T. Cross, Primitive Pythagorean Triples of Gaussian Integers, Mathematics Magazine, 59:4(1986), Leonard E. Dickson, History of the Theory of Numbers, vol. II, Chelsea Publishing, New York, Ernest J. Eckert, The Group of Primitive Pythagorean Triangles, Mathematics Magazine, 57:1(1984), Richard K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer-Verlag, New York, 1994.

14 6. John Leech, The Rational Cubiod Revisited, American Mathematical Monthly, vol. 84 (1977),

### 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

### TRIANGLES ON THE LATTICE OF INTEGERS. Department of Mathematics Rowan University Glassboro, NJ Andrew Roibal and Abdulkadir Hassen

TRIANGLES ON THE LATTICE OF INTEGERS Andrew Roibal and Abdulkadir Hassen Department of Mathematics Rowan University Glassboro, NJ 08028 I. Introduction In this article we will be studying triangles whose

### 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

### 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:

### A Non-Existence Property of Pythagorean Triangles with a 3-D Application

A Non-Existence Property of Pythagorean Triangles with a 3-D Application Konstantine Zelator Department of Mathematics and Computer Science Rhode Island College 600 Mount Pleasant Avenue Providence, RI

### The Equation x 2 + y 2 = z 2

The Equation x 2 + y 2 = z 2 The equation x 2 + y 2 = z 2 is associated with the Pythagorean theorem: In a right triangle the sum of the squares on the sides is equal to the square on the hypotenuse. We

### 4.2 Euclid s Classification of Pythagorean Triples

178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple

### 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

### 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

### Congruent Number Problem

University of Waterloo October 28th, 2015 Number Theory Number theory, can be described as the mathematics of discovering and explaining patterns in numbers. There is nothing in the world which pleases

### k, then n = p2α 1 1 pα k

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

### PERIMETERS OF PRIMITIVE PYTHAGOREAN TRIANGLES. 1. Introduction

PERIMETERS OF PRIMITIVE PYTHAGOREAN TRIANGLES LINDSEY WITCOSKY ADVISOR: DR. RUSS GORDON Abstract. This paper examines two methods of determining whether a positive integer can correspond to the semiperimeter

### 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 =

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### 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

### 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

### 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.

### p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

### 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

### DigitalCommons@University of Nebraska - Lincoln

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University

### Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### 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,

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### 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

### 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,

### Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

### 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

### Practice Problems for First Test

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### 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

### Odd Perfect Numbers. way back to the ancient Greek era. A perfect number is both a positive integer that is the sum of

Stephanie Mijat Math 301 Odd Perfect Numbers Perfect numbers have been an interesting topic of study for mathematicians dating all the way back to the ancient Greek era. A perfect number is both a positive

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

### ARITHMETIC PROGRESSIONS OF FOUR SQUARES

ARITHMETIC PROGRESSIONS OF FOUR SQUARES KEITH CONRAD 1. Introduction Suppose a, b, c, and d are rational numbers such that a 2, b 2, c 2, and d 2 form an arithmetic progression: the differences b 2 a 2,

### 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

### THE CONGRUENT NUMBER PROBLEM

THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

### 2 The Euclidean algorithm

2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range {0,1,...,N

### Topics in Number Theory

Chapter 8 Topics in Number Theory 8.1 The Greatest Common Divisor Preview Activity 1 (The Greatest Common Divisor) 1. Explain what it means to say that a nonzero integer m divides an integer n. Recall

### THE CONGRUENT NUMBER PROBLEM

THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean

### Solutions to Practice Problems

Solutions to Practice Problems March 205. Given n = pq and φ(n = (p (q, we find p and q as the roots of the quadratic equation (x p(x q = x 2 (n φ(n + x + n = 0. The roots are p, q = 2[ n φ(n+ ± (n φ(n+2

### Pythagorean triples. Darryl McCullough. University of Oklahoma

The a, b, c s of Pythagorean triples Darryl McCullough University of Oklahoma September 10, 2001 1 A Pythagorean triple PT is an ordered triple a, b, c of positive integers such that a 2 + b 2 = c 2. When

### 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

### EULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.

EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If

Copyright by Christina Lau 011 The Report Committee for Christina Lau Certifies that this is the approved version of the following report: Pythagorean Theorem Extensions APPROVED BY SUPERVISING COMMITTEE:

### 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

### Algebra for Digital Communication

EPFL - Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain

### 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

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### GREATEST COMMON DIVISOR

DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their

### MIDY S THEOREM FOR PERIODIC DECIMALS. Joseph Lewittes Department of Mathematics and Computer Science, Lehman College (CUNY), Bronx, New York 10468

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A02 MIDY S THEOREM FOR PERIODIC DECIMALS Joseph Lewittes Department of Mathematics and Computer Science, Lehman College (CUNY), Bronx,

### TEXAS A&M UNIVERSITY. Prime Factorization. A History and Discussion. Jason R. Prince. April 4, 2011

TEXAS A&M UNIVERSITY Prime Factorization A History and Discussion Jason R. Prince April 4, 2011 Introduction In this paper we will discuss prime factorization, in particular we will look at some of the

### In a triangle with a right angle, there are 2 legs and the hypotenuse of a triangle.

PROBLEM STATEMENT In a triangle with a right angle, there are legs and the hypotenuse of a triangle. The hypotenuse of a triangle is the side of a right triangle that is opposite the 90 angle. The legs

### 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

### RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES

RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES DARRYL MCCULLOUGH AND ELIZABETH WADE In [9], P. W. Wade and W. R. Wade (no relation to the second author gave a recursion formula that produces Pythagorean

### 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

### Fractions and Decimals

Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### 9 abcd = dcba. 9 ( b + 10c) = c + 10b b + 90c = c + 10b b = 10c.

In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should

### The Laws of Cryptography Cryptographers Favorite Algorithms

2 The Laws of Cryptography Cryptographers Favorite Algorithms 2.1 The Extended Euclidean Algorithm. The previous section introduced the field known as the integers mod p, denoted or. Most of the field

### ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS. 1. Ideals in Commutative Rings In this section all groups and rings will be commutative.

ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS PETE L. CLARK 1. Ideals in Commutative Rings In this section all groups and rings will be commutative. 1.1. Basic definitions and examples. Let R be a (commutative!)

### Number Theory 2. Paul Yiu. Department of Mathematics Florida Atlantic University. Spring 2007 April 8, 2007

Number Theory Paul Yiu Department of Mathematics Florida Atlantic University Spring 007 April 8, 007 Contents 1 Preliminaries 101 1.1 Infinitude of prime numbers............... 101 1. Euclidean algorithm

### Elementary Number Theory

Elementary Number Theory Ahto Buldas December 3, 2016 Ahto Buldas Elementary Number Theory December 3, 2016 1 / 1 Division For any m > 0, we define Z m = {0, 1,... m 1} For any n, m Z (m > 0), there are

### Two-generator numerical semigroups and Fermat and Mersenne numbers

Two-generator numerical semigroups and Fermat and Mersenne numbers Shalom Eliahou and Jorge Ramírez Alfonsín Abstract Given g N, what is the number of numerical semigroups S = a,b in N of genus N\S = g?

### Egyptian fraction representations of 1 with odd denominators

Egyptian fraction representations of with odd denominators P. Shiu Department of Mathematical Sciences Loughborough University Leicestershire LE 3TU United Kingdon Email: P.Shiu@lboro.ac.uk Abstract The

### Stanford University Educational Program for Gifted Youth (EPGY) Number Theory. Dana Paquin, Ph.D.

Stanford University Educational Program for Gifted Youth (EPGY) Dana Paquin, Ph.D. paquin@math.stanford.edu Summer 2010 Note: These lecture notes are adapted from the following sources: 1. Ivan Niven,

### PRINCIPLE OF MATHEMATICAL INDUCTION

Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction Newton was indebted to it for his theorem

### A Study on the Necessary Conditions for Odd Perfect Numbers

A Study on the Necessary Conditions for Odd Perfect Numbers Ben Stevens U63750064 Abstract A collection of all of the known necessary conditions for an odd perfect number to exist, along with brief descriptions

### Pythagorean Theorem: Proof and Applications

Pythagorean Theorem: Proof and Applications Kamel Al-Khaled & Ameen Alawneh Department of Mathematics and Statistics, Jordan University of Science and Technology IRBID 22110, JORDAN E-mail: kamel@just.edu.jo,

### 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

### Oh Yeah? Well, Prove It.

Oh Yeah? Well, Prove It. MT 43A - Abstract Algebra Fall 009 A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### Coding Theory. Kenneth H. Rosen, AT&T Laboratories.

5 Coding Theory Author: Kenneth H. Rosen, AT&T Laboratories. Prerequisites: The prerequisites for this chapter are the basics of logic, set theory, number theory, matrices, and probability. (See Sections

### 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

### MATH 361: NUMBER THEORY FIRST LECTURE

MATH 361: NUMBER THEORY FIRST LECTURE 1. Introduction As a provisional definition, view number theory as the study of the properties of the positive integers, Z + = {1, 2, 3, }. Of particular interest,

### 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

### PYTHAGOREAN NUMBERS. Bowling Green State University, Bowling Green, OH (Submitted February 1988)

S u p r i y a Mohanty a n d S. P. Mohanty Bowling Green State University, Bowling Green, OH 43403-0221 (Submitted February 1988) Let M be a right angled triangle with legs x and y and hypotenuse z. Then

### 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)

### An Innocent Investigation

An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

### 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

### 2. Integers and Algorithms Euclidean Algorithm. Euclidean Algorithm. Suppose a and b are integers

2. INTEGERS AND ALGORITHMS 155 2. Integers and Algorithms 2.1. Euclidean Algorithm. Euclidean Algorithm. Suppose a and b are integers with a b > 0. (1) Apply the division algorithm: a = bq + r, 0 r < b.

### 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

### Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have

### Mathematical induction & Recursion

CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,

### There are 8000 registered voters in Brownsville, and 3 8. of these voters live in

Politics and the political process affect everyone in some way. In local, state or national elections, registered voters make decisions about who will represent them and make choices about various ballot

### Introduction to Diophantine Equations

Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field

### 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

### Grade 6 Math Circles March 24/25, 2015 Pythagorean Theorem Solutions

Faculty of Mathematics Waterloo, Ontario NL 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles March 4/5, 015 Pythagorean Theorem Solutions Triangles: They re Alright When They

### 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

### CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

### 3 Congruence arithmetic

3 Congruence arithmetic 3.1 Congruence mod n As we said before, one of the most basic tasks in number theory is to factor a number a. How do we do this? We start with smaller numbers and see if they divide

### 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

### 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

### 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

### The square-free kernel of x 2n a 2n

ACTA ARITHMETICA 101.2 (2002) The square-free kernel of x 2n a 2n by Paulo Ribenboim (Kingston, Ont.) Dedicated to my long-time friend and collaborator Wayne McDaniel, at the occasion of his retirement

### An Interesting Way to Combine Numbers

An Interesting Way to Combine Numbers Joshua Zucker and Tom Davis November 28, 2007 Abstract This exercise can be used for middle school students and older. The original problem seems almost impossibly

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,