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14. Assume (a, b, c) is a primitive Pythagorean triple with a² + b²c². Fill in these missing steps in the proof of The- orem 5.14. a. Prove that one of a or b must be even and the other must be odd.

14. Assume (a, b, c) is a primitive Pythagorean triple with a² + b²c². Fill in these missing steps in the proof of The- orem 5.14. a. Prove that one of a or b must be even and the other must be odd.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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14. Assume (a, b, c) is a primitive Pythagorean triple with a² + b²c². Fill in these missing steps in the proof of The- orem 5.14. a. Prove that one of a or b must be even and the other must be odd.
Transcribed Image Text:14. Assume (a, b, c) is a primitive Pythagorean triple with a² + b²c². Fill in these missing steps in the proof of The- orem 5.14. a. Prove that one of a or b must be even and the other must be odd.
Theorem 5.14 (Pythagorean Triples Theorem): Any primitive Pythagorean triple (a, b, c) of nat- ural numbers such that a is odd and b is even is given by v² u² 2 v² + u² 2 for suitable odd numbers u and v with v> u. Conversely, for any choice of odd numbers u and v with v> u, the formulas (*) produce a primitive Pythagorean triple (a, b, c) with a odd and beven. (*) a = uv; b = -Public key network security systems There are also important practical applications of the Fundamental Theorem of Arith- metic outside of the theory of numbers. Public key encoding is a widely used secu- rity system that is designed to assure communication privacy in computer networks. The system relies on pairs of keys (or passwords) that determine key-specific schemes that are used to encode and decode messages. Each pair of keys consists of a public key, which is usually made available to everyone using the network, and a private key, which is known only to the person who has been assigned that pair. Anyone who knows the public key can encode (encrypt) a message and send it to another person on the network, but only that other person knows the private key that can decode (decrypt) the message. This makes it possible to exchange messages securely over an insecure network. Public and private keys are produced in pairs by computer programs. The most well-known of these is the RSA public key encryption algorithm, which is based on prime factorization and modular arithmetic (see Unit 6.1). The RSA algorithm derives its name from the first letters of the last names of its inventors, Ronald L. Rivest, Adi Shamir, and Leonard M. Adleman, who jointly published a six-page paper in 1978 that described the system. The RSA algorithm takes two very large prime numbers, P and Q, and forms their product to obtain a very large composite number N = PQ. (N may have 150 digits or so.) The RSA algorithm then generates an "exponent" E, and an "inverse exponent" D, so that E and the number (P-1)(Q-1) are relatively prime. The public key consists of E and N while D and N comprise the corresponding private key. If someone knows the public key and also knows the two prime factors P and Q of N, it would not be difficult for them to compute the corresponding private key. However, factoring the number N into its large prime factors P and Q is virtually impossible because there are no known computer factorization algorithms that are efficient enough to find these factorizations in any reasonable length of time. More- over, the Fundamental Theorem of Arithmetic assures that no simpler factorization of N is possible. Thus, the task of computing the private key from a public key is vir- PQ. tually impossible without knowing the prime factorization N ==
Transcribed Image Text:Theorem 5.14 (Pythagorean Triples Theorem): Any primitive Pythagorean triple (a, b, c) of nat- ural numbers such that a is odd and b is even is given by v² u² 2 v² + u² 2 for suitable odd numbers u and v with v> u. Conversely, for any choice of odd numbers u and v with v> u, the formulas (*) produce a primitive Pythagorean triple (a, b, c) with a odd and beven. (*) a = uv; b = -Public key network security systems There are also important practical applications of the Fundamental Theorem of Arith- metic outside of the theory of numbers. Public key encoding is a widely used secu- rity system that is designed to assure communication privacy in computer networks. The system relies on pairs of keys (or passwords) that determine key-specific schemes that are used to encode and decode messages. Each pair of keys consists of a public key, which is usually made available to everyone using the network, and a private key, which is known only to the person who has been assigned that pair. Anyone who knows the public key can encode (encrypt) a message and send it to another person on the network, but only that other person knows the private key that can decode (decrypt) the message. This makes it possible to exchange messages securely over an insecure network. Public and private keys are produced in pairs by computer programs. The most well-known of these is the RSA public key encryption algorithm, which is based on prime factorization and modular arithmetic (see Unit 6.1). The RSA algorithm derives its name from the first letters of the last names of its inventors, Ronald L. Rivest, Adi Shamir, and Leonard M. Adleman, who jointly published a six-page paper in 1978 that described the system. The RSA algorithm takes two very large prime numbers, P and Q, and forms their product to obtain a very large composite number N = PQ. (N may have 150 digits or so.) The RSA algorithm then generates an "exponent" E, and an "inverse exponent" D, so that E and the number (P-1)(Q-1) are relatively prime. The public key consists of E and N while D and N comprise the corresponding private key. If someone knows the public key and also knows the two prime factors P and Q of N, it would not be difficult for them to compute the corresponding private key. However, factoring the number N into its large prime factors P and Q is virtually impossible because there are no known computer factorization algorithms that are efficient enough to find these factorizations in any reasonable length of time. More- over, the Fundamental Theorem of Arithmetic assures that no simpler factorization of N is possible. Thus, the task of computing the private key from a public key is vir- PQ. tually impossible without knowing the prime factorization N ==
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