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Co 2. (10 points) Let (a, b, c) be a primitive Pythagorean triple. Prove that cis...
2. The triple (a, b, c) is called a Pythagorean triple if a, b and c...
2. The triple (a, b, c) is called a Pythagorean triple if a, b and c are natural numbers (in other words, they belong to N) and a2 +bc2. Prove that if a rectangle has sides of rational lengths p and q then the length of its diagonal is irrational if and only if there is a natural number r such that (p,q,r) is a a Pythagorean triple. State clearly any theorem proved in class that you use.
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The prec...
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The preceding theorem lets us easily generale infinitely many Pythagorean triples, but, in fact, cvery primitive Pythagorean triple can be generated by chousing appropriale natural numbers s and and making the Pythagorean triple as described in the preceding thcorem. As a hint to the proof, we make a little observation. 8.6 Lemma. Let (a, b,e) be a primitive Pythagorean...
Discrete Structures class. An ordered triple of positive integers (a,b,c) is called a Pythagorean Triple if...
Discrete Structures class.
An ordered triple of positive integers (a,b,c) is called a
Pythagorean Triple if a^2 + b^2 = c^2. Prove that if m & n are
pos int....
2) (10 pts) An ordered triple of positive integers (a, b, c) is called a Pythagorean Triple if a² + b2 = c2. Prove that if m and n are positive integers with m > n, then (m? – nº, 2mn, m² + n°) is a Pythagorean triple. Use this...
Recall we say that a Pythagorean triple is a triplet of positive integers a, b and c such that a^...
Recall we say that a Pythagorean triple is a triplet of positive integers a, b and c such that a^2 + b^2 = c^2. Examples are (3,4,5) and (5,12,13). Explain why for any Pythagorean triplet one must have that 12 divides abc. (Hint: It may be easiest to do this by showing that 3 divides abc and showing separately that 4 divides abc).
Re doing. A Pythagorean triple is a set of integers (a, b, c) satisfying a2+bc2. Write pseudocode...
re doing. A Pythagorean triple is a set of integers (a, b, c) satisfying a2+bc2. Write pseudocode (or Matlab code) that will output the number of unique Pythagorean triples that satisfy c< 200. [Unique means that you should not count (a 3, b 4, c 5) and (a 4, b 3, c 5) as two different triples, for example]. Explain why your program will work. Imagine a square n x n matrix A with diagonal elements which we believe to...
10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b...
10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime. han C: oadl Prove that if p, q, and r are three different prime numbers, then p2 + q2 #r2 11.
6. (10 points) Let A, B, and C be sets. Prove (AuB)C(AnC) u(BnC)
6. (10 points) Let A, B, and C be sets. Prove (AuB)C(AnC) u(BnC)
Let p be an odd prime. Prove that if g is a primitive root modulo p, then g^(p-1)/2 ≡ -1 (mod p).
Let
p be an odd prime. Prove that if g is a primitive root modulo p,
then g^(p-1)/2 ≡ -1 (mod p).
Let p be an odd prime. Prove that if g is a primitive root modulo p, then go-1)/2 =-1 (mod p) Hint: Use Lemma 2 from Chapter 28 (If p is prime and d(p 1), then cd-1 Ξ 0 (mod p) has exactly d solutions).
Let p be an odd prime. Prove that if g is a primitive...
6. (16 points) Let CE C be a primitive n-th root of unity. Let X =...
6. (16 points) Let CE C be a primitive n-th root of unity. Let X = 6 + 1/5. (a) (4 points) Show that Q(5) R = Q(1). (b) (4 points) Let f be the minimal polynomial of over Q. Show that Q(x) is a splitting field of f over Q. (c) (4 points) Show that Gal(Q(^)/Q) – (Z/nZ)* / (-1). (d) (4 points) Find the minimal polynomial of 2 cos(27/9) over Q.
Computing another Galois Group a) Let? = eri/6 be a primitive 12th root of unity. Prove...
Computing another Galois Group a) Let? = eri/6 be a primitive 12th root of unity. Prove that is a zero of the polynomial t4 - t+ 1, and that the other zeros are 55,57,511. b) Prove that t4 – +2 + 1 is irreducible over Q and is the minimal polynomial of over Q.
2. The triple (a, b, c) is called a Pythagorean triple if a, b and c are natural numbers (in other words, they belong to N) and a2 +bc2. Prove that if a rectangle has sides of rational lengths p and q then the length of its diagonal is irrational if and only if there is a natural number r such that (p,q,r) is a a Pythagorean triple. State clearly any theorem proved in class that you use.
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The preceding theorem lets us easily generale infinitely many Pythagorean triples, but, in fact, cvery primitive Pythagorean triple can be generated by chousing appropriale natural numbers s and and making the Pythagorean triple as described in the preceding thcorem. As a hint to the proof, we make a little observation. 8.6 Lemma. Let (a, b,e) be a primitive Pythagorean...
Discrete Structures class.
An ordered triple of positive integers (a,b,c) is called a
Pythagorean Triple if a^2 + b^2 = c^2. Prove that if m & n are
pos int....
2) (10 pts) An ordered triple of positive integers (a, b, c) is called a Pythagorean Triple if a² + b2 = c2. Prove that if m and n are positive integers with m > n, then (m? – nº, 2mn, m² + n°) is a Pythagorean triple. Use this...
re doing. A Pythagorean triple is a set of integers (a, b, c) satisfying a2+bc2. Write pseudocode (or Matlab code) that will output the number of unique Pythagorean triples that satisfy c< 200. [Unique means that you should not count (a 3, b 4, c 5) and (a 4, b 3, c 5) as two different triples, for example]. Explain why your program will work. Imagine a square n x n matrix A with diagonal elements which we believe to...
10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime. han C: oadl Prove that if p, q, and r are three different prime numbers, then p2 + q2 #r2 11.
6. (10 points) Let A, B, and C be sets. Prove (AuB)C(AnC) u(BnC)
Let
p be an odd prime. Prove that if g is a primitive root modulo p,
then g^(p-1)/2 ≡ -1 (mod p).
Let p be an odd prime. Prove that if g is a primitive root modulo p, then go-1)/2 =-1 (mod p) Hint: Use Lemma 2 from Chapter 28 (If p is prime and d(p 1), then cd-1 Ξ 0 (mod p) has exactly d solutions).
Let p be an odd prime. Prove that if g is a primitive...
6. (16 points) Let CE C be a primitive n-th root of unity. Let X = 6 + 1/5. (a) (4 points) Show that Q(5) R = Q(1). (b) (4 points) Let f be the minimal polynomial of over Q. Show that Q(x) is a splitting field of f over Q. (c) (4 points) Show that Gal(Q(^)/Q) – (Z/nZ)* / (-1). (d) (4 points) Find the minimal polynomial of 2 cos(27/9) over Q.
Computing another Galois Group a) Let? = eri/6 be a primitive 12th root of unity. Prove that is a zero of the polynomial t4 - t+ 1, and that the other zeros are 55,57,511. b) Prove that t4 – +2 + 1 is irreducible over Q and is the minimal polynomial of over Q.
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