09-240:HW2 - Revision history

C8sd: 117,648 = 7^6 - 1

2009-12-07T03:18:03Z

117,648 = 7^6 - 1

← Older revision		Revision as of 23:18, 6 December 2009
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	* Problems <u>3a</u> and 3bcd on page 6, problems <u>1</u>, 7, <u>18</u>, 19 and <u>21</u> on pages 14-16 and problems <u>8</u>, 9, 11 and <u>19</u> on pages 20-21. You need to submit only the underlined problems.		* Problems <u>3a</u> and 3bcd on page 6, problems <u>1</u>, 7, <u>18</u>, 19 and <u>21</u> on pages 14-16 and problems <u>8</u>, 9, 11 and <u>19</u> on pages 20-21. You need to submit only the underlined problems.

	* Note that the numbers <math>1^6-1=0</math>, <math>2^6-1=63</math>, <math>3^6-1=728</math>, <math>4^6-1=4,095</math>, <math>5^6-1=15,624</math> and <math>6^6-1=~~117~~,~~648~~</math> are all divisible by <math>7</math>. The following four part exercise explains that this is not a coincidence. But first, let <math>p</math> be some odd prime number and let <math>{\mathbb F}_p</math> be the field with p elements as defined in class.		* Note that the numbers <math>1^6-1=0</math>, <math>2^6-1=63</math>, <math>3^6-1=728</math>, <math>4^6-1=4,095</math>, <math>5^6-1=15,624</math> and <math>6^6-1=46,655</math> are all divisible by <math>7</math>. The following four part exercise explains that this is not a coincidence. But first, let <math>p</math> be some odd prime number and let <math>{\mathbb F}_p</math> be the field with p elements as defined in class.
	*# Prove that the product <math>b:=1\cdot 2\cdot\ldots\cdot(p-2)\cdot(p-1)</math> is a non-zero element of <math>{\mathbb F}_p</math>.		*# Prove that the product <math>b:=1\cdot 2\cdot\ldots\cdot(p-2)\cdot(p-1)</math> is a non-zero element of <math>{\mathbb F}_p</math>.
	*# Let <math>a</math> be a non-zero element of <math>{\mathbb F}_p</math>. Prove that the sets <math>\{1,2,\ldots,(p-1)\}</math> and <math>\{1a,2a,\ldots,(p-1)a\}</math> are the same (though their elements may be listed here in a different order).		*# Let <math>a</math> be a non-zero element of <math>{\mathbb F}_p</math>. Prove that the sets <math>\{1,2,\ldots,(p-1)\}</math> and <math>\{1a,2a,\ldots,(p-1)a\}</math> are the same (though their elements may be listed here in a different order).

Drorbn at 22:14, 23 September 2009

2009-09-23T22:14:13Z

← Older revision		Revision as of 18:14, 23 September 2009
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	{{09-240/Navigation}}		{{09-240/Navigation}}
	{{In Preparation}}

	Read sections 1.1 through 1.3 in our textbook, and solve the following problems:		Read sections 1.1 through 1.3 in our textbook, and solve the following problems:

Drorbn at 21:44, 22 September 2009

2009-09-22T21:44:08Z

← Older revision		Revision as of 17:44, 22 September 2009
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	You don't need to submit this exercise at all, but you will learn a lot by doing it!		You don't need to submit this exercise at all, but you will learn a lot by doing it!

	This assignment is due at the tutorials on Thursday ~~September~~ 28. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorials on Thursday October 1. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn at 22:55, 21 September 2009

2009-09-21T22:55:12Z

New page

{{09-240/Navigation}}
{{In Preparation}}

Read sections 1.1 through 1.3 in our textbook, and solve the following problems:

* Problems 3a and 3bcd on page 6, problems 1, 7, 18, 19 and 21 on pages 14-16 and problems 8, 9, 11 and 19 on pages 20-21. You need to submit only the underlined problems.

* Note that the numbers <math>1^6-1=0</math>, <math>2^6-1=63</math>, <math>3^6-1=728</math>, <math>4^6-1=4,095</math>, <math>5^6-1=15,624</math> and <math>6^6-1=117,648</math> are all divisible by <math>7</math>. The following four part exercise explains that this is not a coincidence. But first, let <math>p</math> be some odd prime number and let <math>{\mathbb F}_p</math> be the field with p elements as defined in class.
*# Prove that the product <math>b:=1\cdot 2\cdot\ldots\cdot(p-2)\cdot(p-1)</math> is a non-zero element of <math>{\mathbb F}_p</math>.
*# Let <math>a</math> be a non-zero element of <math>{\mathbb F}_p</math>. Prove that the sets <math>\{1,2,\ldots,(p-1)\}</math> and <math>\{1a,2a,\ldots,(p-1)a\}</math> are the same (though their elements may be listed here in a different order).
*# With <math>a</math> and <math>b</math> as in the previous two parts, show that <math>ba^{p-1}=b</math> in <math>{\mathbb F}_p</math>, and therefore <math>a^{p-1}=1</math> in <math>{\mathbb F}_p</math>.
*# How does this explain the fact that <math>4^6-1</math> is divisible by <math>7</math>?
You don't need to submit this exercise at all, but you will learn a lot by doing it!

This assignment is due at the tutorials on Thursday September 28. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.