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April 8th, 2014, 02:49 AM  #1 
Senior Member Joined: Oct 2013 From: Far far away Posts: 429 Thanks: 18  quadratic function
Prove that there is a quadratic function f(n) = n^2 + bn + c with positive integer coefficients b, c such that f(n) is composite (i.e. not prime) for all positive integers n, or else prove that the statement is false. I don't even know where to start so any help will be deeply appreciated. thanks 
April 8th, 2014, 03:58 AM  #2 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361 
A quadratic function is composite if you can factor it in at least two polynomials each different from one with integer coefficients. To prove that there is a quadratic function f(n) = n^2 + bn + c with positive integer coefficients b, c such that f(n) is composite (i.e. not prime) for all positive integers n, you just need to find an example. A general quadratic function (with respect to n) is an^2 + bn + c But in your case, a = 1 so you cannot factor out a constant. Another lower degree polynomial is of the form wn + x, of which there are two factors. Can you find two polynomials of the form wn + x such that when multiplied you get such a function? 

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