There are often choices along the way about how to break a number down into the...
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There are often choices along the way about how to break a number down into the sum of two smaller numbers. The next sum-product table starts from 4 going to 2+2 instead of 3 + 1. 4= (2+2) 2.2 = 4 = (1+1) +2 1.1=1 = 1+1+(1+1) 1.1=1 6 Curiously, the sum of the products is again 6. (a) Start from 5 and make 3 different sum-product tables (realizing that 1+ 4 versus 4+1 isn't really different). (b) Pick n with n with 7 ≤ n ≤ 10 and make at least 5 different sum- product tables starting from n. (c) Make a conjecture about the sum of the products in these tables when you start with a positive integer n ≥ 2 (your answer should involve n). 3. Primes from polynomials Recall that a prime number has no factors besides 1 and itself. A composite number is not prime, i.e., it has factors besides 1 and itself. For large numbers, you can use an online resource like Wolfram Alpha to factor numbers or confirm that they are prime. (a) Consider the polynomial g(n) = 2n² +29. Find at least 10 different natural numbers n such that g(n) is a prime number (don't just use 0, 1,... 9). Find at least 2 different natural numbers such that g(n) is composite. (b) Repeat for the polynomial h(n) = n²n + 41. There are often choices along the way about how to break a number down into the sum of two smaller numbers. The next sum-product table starts from 4 going to 2+2 instead of 3 + 1. 4= (2+2) 2.2 = 4 = (1+1) +2 1.1=1 = 1+1+(1+1) 1.1=1 6 Curiously, the sum of the products is again 6. (a) Start from 5 and make 3 different sum-product tables (realizing that 1+ 4 versus 4+1 isn't really different). (b) Pick n with n with 7 ≤ n ≤ 10 and make at least 5 different sum- product tables starting from n. (c) Make a conjecture about the sum of the products in these tables when you start with a positive integer n ≥ 2 (your answer should involve n). 3. Primes from polynomials Recall that a prime number has no factors besides 1 and itself. A composite number is not prime, i.e., it has factors besides 1 and itself. For large numbers, you can use an online resource like Wolfram Alpha to factor numbers or confirm that they are prime. (a) Consider the polynomial g(n) = 2n² +29. Find at least 10 different natural numbers n such that g(n) is a prime number (don't just use 0, 1,... 9). Find at least 2 different natural numbers such that g(n) is composite. (b) Repeat for the polynomial h(n) = n²n + 41.
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a Starting from 5 1 5 2 3 2 3 6 2 5 1 4 1 4 4 3 5 1 1 3 1 1 3 3 b For n 7 1 7 3 4 3 4 12 2 7 1 6 1 6 ... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
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