Vipul: Created page with "==Statement== Suppose $p$ is a polynomial with integer coefficients, i.e., $p(x) \in \mathbb{Z}[x]$ . Suppose that all the coefficients of $p$ ..."

2012-07-03T00:52:41Z

Created page with "==Statement== Suppose <math>p</math> is a polynomial with integer coefficients, i.e., <math>p(x) \in \mathbb{Z}[x]</math>. Suppose that all the coefficients of <math>p</math>..."

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==Statement==

Suppose <math>p</math> is a polynomial with integer coefficients, i.e., <math>p(x) \in \mathbb{Z}[x]</math>. Suppose that all the coefficients of <math>p</math> are ''nonnegative''. Further, suppose <math>b \ge 2</math> is a natural number strictly greater than all coefficients. Then, if <math>p(b)</math> is a [[prime number]], <math>p</math> must be an [[irreducible polynomial]].

An alternate formulation is as follows: for any <math>b \ge 2</math>, if a number with digits <math>a_na_{n-1} \dots a_1a_0</math> written in base <math>b</math> is prime (so in particular <math>0 \le a_i \le b - 1</math> for <math>i \in \{ 0,1,\dots,n\}</math>) then the polynomial <math>a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0</math> is irreducible.

==Related facts==

* [[Bunyakovsky conjecture]] is a conjectured converse of sorts: if a polynomial is irreducible and the set of its values does not have a gcd, then the polynomial must take prime values at infinitely many natural numbers.

Cohn's irreducibility criterion - Revision history

Vipul: Created page with "==Statement== Suppose p is a polynomial with integer coefficients, i.e., p(x) \in \mathbb{Z}[x]. Suppose that all the coefficients of p..."

Vipul: Created page with "==Statement== Suppose $p$ is a polynomial with integer coefficients, i.e., $p(x) \in \mathbb{Z}[x]$ . Suppose that all the coefficients of $p$ ..."