Fundamental theorem of arithmetic: Difference between revisions

Revision as of 11:44, 27 April 2014

Revision as of 13:44, 27 April 2014

Line 14:

==History==

Book VII, propositions 30 and 32 of [[Euclid]]'s [[Euclid's Elements|Elements]] is essentially the statement and proof of the fundamental theorem. Article 16 of [[Carl Friedrich Gauss|Gauss]]' ''[[Disquisitiones Arithmeticae]]'' is an early modern statement and proof employing [[modular arithmetic]].<ref name="Gauss1801.loc=16" />

{{Quotation|

If two numbers by multiplying one another make some

number, and any prime number measure the product, it will

also measure one of the original numbers.

|Euclid|[[#CITEREFEuclidHeath1956|Elements VII]], Proposition 30}}

The proposition 30 is refered to [[Euclid's lemma]]. And it is the key in the proof of the fundamental theorem of arithmetic.

{{Quotation|

Any composite number is measured by some prime number.

|Euclid|[[#CITEREFEuclidHeath1956|Elements VII]], Proposition 31}}

The proposition 31 is derived from the proposition 30.

{{Quotation|

Any number either is prime or is measured by some prime number.

|Euclid|[[#CITEREFEuclidHeath1956|Elements VII]], Proposition 32}}

The proposition 32 is derived from the proposition 31.

Article 16 of [[Carl Friedrich Gauss|Gauss]]' ''[[Disquisitiones Arithmeticae]]'' is an early modern statement and proof employing [[modular arithmetic]].<ref name="Gauss1801.loc=16" />

==Applications==

Line 227 ⟶ 249:

| publisher=Cambridge University Press

| isbn=978-0-521-28654-1

}}

* {{citation

| author1 = Euclid

| author1-link = Euclid

| last2 = Heath | first2 = Thomas L. (translator into English)

| author2-link = Thomas Little Heath

| title = The thirteen books of the Elements

| edition = Second Edition Unabridged

| volume = 2 (Books III-IX)

| publisher = [[Dover]]

| ___location = New York

| year = 1956

| isbn = 978-0-486-60089-5

| url = http://store.doverpublications.com/0486600890.html

}}

* {{Citation