tag:blogger.com,1999:blog-2400513859305780710.post3695134078906077665..comments2023-04-04T07:53:53.789-07:00Comments on Mathematical Food For Thought: Why Do You Exist? Topic: Number Theory. Level: AIME.Jeffrey Wanghttp://www.blogger.com/profile/11114458640271201663noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-2400513859305780710.post-49978637451511795402006-08-15T16:09:08.000-07:002006-08-15T16:09:08.000-07:00We can assume WLOG that (x, y) = 1, since the equa...We can assume WLOG that (x, y) = 1, since the equation is homogeneous and we can divide out by the gcd.<br><br>But then y | x^2 and x | y^2. For x, y > 1 this implies common factors. Then x = y = 1, giving k = 2 as expected.QCnoreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-2348234577954187862006-08-16T02:49:43.000-07:002006-08-16T02:49:43.000-07:00We can also take the discriminant as a quadratic i...We can also take the discriminant as a quadratic in x and get that (k^2-4)y^2 must be a square, so k = 2.paladin8noreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-83546440287380662472006-08-16T12:34:28.000-07:002006-08-16T12:34:28.000-07:00"But 3n-1 \equiv 2 \pmod{3} , which is not a ..."But 3n-1 \equiv 2 \pmod{3} , which is not a quadratic residue modulo 3 so there are no solutions"<br><br>That's so complicated-sounding. Why don't you just say that 2 isn't a square mod 3 to be simple?Moonoreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-27844670835030358882006-08-16T12:50:38.000-07:002006-08-16T12:50:38.000-07:00Well, square mod 3 isn't really well defined.....Well, square mod 3 isn't really well defined... on the other hand, quadratic residue is.paladin8noreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-60307232247001271842006-08-16T15:17:18.000-07:002006-08-16T15:17:18.000-07:00All the cool PROMYS kids say quadratic residue :)All the cool PROMYS kids say quadratic residue :)QCnoreply@blogger.com