SOLUTION: Let P(x) be a nonconstant polynomial, where all the coefficients are nonnegative integers. Prove that there exist infinitely many positive integers n such that P(n) is composite. I

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Let P(x) be a nonconstant polynomial, where all the coefficients are nonnegative integers. Prove that there exist infinitely many positive integers n such that P(n) is composite. I      Log On


   

Question 1173733: Let P(x) be a nonconstant polynomial, where all the coefficients are nonnegative integers. Prove that there exist infinitely many positive integers n such that P(n) is composite. I know that this was already solved on the forum, but I'm curious to know if there is another way to solve it. Thank you!
Answer by ikleyn(52832) About Me  (Show Source):
You can put this solution on YOUR website!
.

                                        For your info:


I found  (thanks to  GOOGLE)  that post in wwww.algebra.com
to which you missed to refer,  relevant to this problem.

https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Polynomials-and-rational-expressions.faq.question.1160694.html

I read that solution and came to the conclusion that it is  INCORRECT,  unfortunately.


            That solution would work for polynomials with the constant term different from  1,

            but  DOES  NOT  work for polynomials with the constant term equal to  1.


            From the other side,  only such polynomials with the constant term  1  are of interest ---
            --- all the others are  OUT  OF  the  INTEREST,  because for them the statement is  TRIVIAL.


I also looked with this request to popular  "people"  web-sites,  and noticed,  that they repeat the same error.


/\/\/\/\/\/\/\/


After some time break, I browsed in the Internet again, looking for appropriate sources.


I found this article online

http://www.cs.umd.edu/~gasarch/BLOGPAPERS/polyprimes.pdf

where the statement is proved 

    Let f(x) ∈ Z[x]. There are an infinite number of y ∈ Z such that f(y) is not prime.

See Colollary 2.2 in this article.