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homework3

# homework3 - a)7,11 b 101,303 c 1331 5005 d 5040 7700 3 Show...

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THEORY OF NUMBERS, Math 115 A Homework 3 Due Friday October 19 1. Consider the “number system” { a + b - 5 : a, b Z } (as a subset of the complex numbers. That is, - 5 = 5 i ). It is usually denoted Z [ - 5]. We deﬁne the “squared norm” N ( a + b - 5) = a 2 + 5 b 2 , which is a non- negative integer. (a) Show that for any two numbers m and n in this system one has N ( mn ) = N ( m ) N ( n ). (b) Write down all the numbers m with N ( m ) 10. (c) A number p is prime in this system if the only factorizations p = mn of it are those in which one (or both) of n and m is in { 1 , - 1 } . Show that the following numbers are primes: 1 + - 5 , 1 - - 5 , 2 , 3 . (d) Show that 5 is not a prime, in this number system. Show that 6 is not a prime and, actually, it has two diﬀerent factorizations. 2. Find the least common multiple of each of the following pairs of integers:
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Unformatted text preview: a)7,11 b) 101,303 c) 1331, 5005 d) 5040, 7700. 3. Show that √ 2 + √ 3 is irrational, by ﬁnding a monic polynomial of which it is the root. 4. Prove that there are inﬁnitely many primes of the form 6 k + 5, where k is a positive integer. 5. Using the Fermat factorization method, factor each of the following posi-tive integers: 73, 8051, 46009, 3200399. 6. Factoring kn by the Fermat factorization method, where k is a small positive integer, is sometimes easier than factoring n by this method. For example, show it is easier to factor 2703 than to factor 901. 1...
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