101: Difference between revisions

Latest revision as of 18:23, 3 July 2012

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Summary

Factorization

The number 101 is a prime number.

Polynomials

Prime-generating polynomials

Below are some polynomials that give prime numbers for small input values, which give the value 101 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value $n$ at which it generates 101
$n^{2}-n+11$	2	all numbers 1-10, because 11 is one of the lucky numbers of Euler.	10
$2n^{2}+29$	2	all numbers 0-28	6

Irreducible polynomials by Cohn's irreducibility criterion

By Cohn's irreducibility criterion, we know that if we write 101 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here the irreducible polynomials of this type that are of degree at least two:

Base $b$	101 in base $b$	Corresponding irreducible polynomial
2	1100101	$x^{6}+x^{5}+x^{2}+1$
3	10202	$x^{4}+2x^{2}+2$
4	1211	$x^{3}+2x^{2}+x+1$
5	401	$4x^{2}+1$
6	245	$2x^{2}+4x+5$
7	203	$2x^{2}+3$
8	145	$x^{2}+4x+5$
9	122	$x^{2}+2x+2$
10	101	$x^{2}+1$

@@ Line 7: / Line 7: @@
 The number 101 is a [[prime number]].
-==Prime-generating polynomials==
+==Polynomials==
+===Prime-generating polynomials===
 Below are some polynomials that give prime numbers for small input values, which give the value 101 for suitable input choice.
@@ Line 17: / Line 19: @@
 |-
 | <math>2n^2 + 29</math> || 2 || all numbers 0-28 || 6
+|}
+===Irreducible polynomials by Cohn's irreducibility criterion===
+By [[Cohn's irreducibility criterion]], we know that if we write 101 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here the irreducible polynomials of this type that are of degree at least two:
+{| class="sortable" border="1"
+! Base <math>b</math> !! 101 in base <math>b</math> !! Corresponding irreducible polynomial
+|-
+| 2 || 1100101 || <math>x^6 + x^5 + x^2 + 1</math>
+|-
+| 3 || 10202 || <math>x^4 + 2x^2 + 2</math>
+|-
+| 4 || 1211 || <math>x^3 + 2x^2 + x + 1</math>
+|-
+| 5 || 401 || <math>4x^2 + 1</math>
+|-
+| 6 || 245 || <math>2x^2 + 4x + 5</math>
+|-
+| 7 || 203 || <math>2x^2 + 3</math>
+|-
+| 8 || 145 || <math>x^2 + 4x + 5</math>
+|-
+| 9 || 122 || <math>x^2 + 2x + 2</math>
+|-
+| 10 || 101 || <math>x^2 + 1</math>
 |}