Vipul at 21:33, 29 January 2014

2014-01-29T21:33:59Z

← Older revision		Revision as of 21:33, 29 January 2014
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	* There is a [[prime number]] <math>p</math> such that the product <math>\prod_{i=1}^k f_i(x)</math> is <math>p</math> times an integer-valued polynomial. In other words, one of the polynomials <math>f_i(x)</math> is always congruent to 1 modulo <math>p</math>.		* There is a [[prime number]] <math>p</math> such that the product <math>\prod_{i=1}^k f_i(x)</math> is <math>p</math> times an integer-valued polynomial. In other words, one of the polynomials <math>f_i(x)</math> is always congruent to 1 modulo <math>p</math>.
	* There exist infinitely many [[natural number]s <math>n</math> for which ''all'' the values <math>f_i(n)</math> are ''simultaneously'' [[prime number\|prime]].		* There exist infinitely many [[natural number]s <math>n</math> for which ''all'' the values <math>f_i(n)</math> are ''simultaneously'' [[prime number\|prime]].

			==Related facts and conjectures==

			===Stronger facts and conjectures===

			* [[Schinzel's hypothesis H]] generalizes from linear polynomials to polynomial of arbitrary degree.
			* [[Bateman-Horn conjecture]] further generalies Schinzel's hypothesis H by providing an asymptotic quantitative estimate of the frequency of occurrence of primes.

			===Weaker facts and conjectures===

			* [[Green-Tao theorem]]
			* [[Twin prime conjecture]]
			* [[Polignac's conjecture]]

Vipul: Created page with "==Statement== Suppose $a_1,a_2,\dots,a_k,b_1,b_2,\dots,b_k$ are integers with all the $a_i \ge 1$ . Then, consider the polynomials: $f_i(x) := a_ix..."$

2014-01-29T21:29:45Z

Created page with "==Statement== Suppose <math>a_1,a_2,\dots,a_k,b_1,b_2,\dots,b_k</math> are integers with all the <math>a_i \ge 1</math>. Then, consider the polynomials: <math>f_i(x) := a_ix..."

New page

==Statement==

Suppose <math>a_1,a_2,\dots,a_k,b_1,b_2,\dots,b_k</math> are integers with all the <math>a_i \ge 1</math>. Then, consider the polynomials:

<math>f_i(x) := a_ix + b_i, i \in \{ 1,2,\dots,n \}</math>

Then, one of the following is true:

* There is a [[prime number]] <math>p</math> such that the product <math>\prod_{i=1}^k f_i(x)</math> is <math>p</math> times an integer-valued polynomial. In other words, one of the polynomials <math>f_i(x)</math> is always congruent to 1 modulo <math>p</math>.
* There exist infinitely many [[natural number]s <math>n</math> for which ''all'' the values <math>f_i(n)</math> are ''simultaneously'' [[prime number|prime]].

Dickson's conjecture - Revision history

Vipul at 21:33, 29 January 2014

Vipul: Created page with "==Statement== Suppose a_1,a_2,\dots,a_k,b_1,b_2,\dots,b_k are integers with all the a_i \ge 1. Then, consider the polynomials: f_i(x) := a_ix..."

Vipul: Created page with "==Statement== Suppose $a_1,a_2,\dots,a_k,b_1,b_2,\dots,b_k$ are integers with all the $a_i \ge 1$ . Then, consider the polynomials: $f_i(x) := a_ix..."$