Changes

==Typical examples==

==Typical examples==

===Euler's conjecture===

===Euler's conjecture===

Based on evidence from unsuccessful manual searches of relatively few numbers, Euler claimed that there were no [[whole number]] solutions to the following equation, similar to one pertaining to famous [[Fermat's Last Theorem]]:

Based on evidence from unsuccessful manual searches of relatively few numbers, [[Euler]] predicted that there were no [[whole number]] solutions to the following equation, similar to one pertaining to famous [[Fermat's Last Theorem]]:

:<math>x^4 + y^4 + z^4 = w^4</math>  

:<math>x^4 + y^4 + z^4 = w^4</math>  

For two hundred years nobody could disprove this claim despite years of computer sifting. Lack of a [[counter-example]] was interpreted as strong evidence in favour of a theory until Naom Elkies of Harvard University discovered the solution in 1988. Despite all the evidence, Euler's conjecture turned out to be false at the end. Extrapolating a theory to cover an infinity of numbers based on insufficient and limited amount of evidence without absolute proof has shown to be an unacceptable gamble. The moral is that it is not possible to use evidence from first local set of million numbers to prove the theory or rather conjecture about global set of all numbers.<ref>{{cite book |title=Fermat's Last Theorem |author=Simon Singh |publisher=Fourth Estate |place=London |year=1997 |pages= 177-178|url=http://books.google.no/books?id=Ncrnn9hCn_kC&dq=simon+singh&hl=en&sa=X&ei=29NyUrXWEIGv4ASqzIDoAQ&redir_esc=y |isbn=1-85702-521-0}}</ref>

For two hundred years nobody could disprove this claim despite years of computer sifting. Lack of a [[counter-example]] was interpreted as strong evidence in favor of a theory until Naom Elkies of Harvard University discovered the solution in 1988.<ref>{{cite book |title=The True Marvel of Numbers: And How Fermat Proved His Last Theorem! |author=David Searle |publisher=AuthorHouse |year=2009 |pages=71 |url=http://books.google.sk/books?id=PQC8AQiFCSsC&pg=PA71&lpg=PA71&dq=Naom+Elkies+of+Harvard+University+1988&source=bl&ots=vtJbAgMJdI&sig=rGHNn8o_kiBJ9Vx9o5XpP41Ep2k&hl=en&sa=X&ei=8OBzUuPcLKi24ATAjYGwCw&ved=0CFYQ6AEwBQ#v=onepage&q=Naom%20Elkies%20of%20Harvard%20University%201988&f=false |isbn=978-1-4389-4530-9}}</ref> Despite all the evidence, Euler's conjecture turned out to be false at the end. Extrapolating a theory to cover an infinity of numbers based on insufficient and limited amount of [[scientific evidence|evidence]] without absolute proof has shown to be an unacceptable gamble. The moral is that it is not possible to use evidence from first local set of million numbers to prove the theory or rather conjecture about global set of all numbers.<ref>{{cite book |title=Fermat's Last Theorem |author=Simon Singh |publisher=Fourth Estate |place=London |year=1997 |pages= 177-178|url=http://books.google.no/books?id=Ncrnn9hCn_kC&dq=simon+singh&hl=en&sa=X&ei=29NyUrXWEIGv4ASqzIDoAQ&redir_esc=y |isbn=1-85702-521-0}}</ref>

== See Also ==

== See Also ==