回复：Thanks for your Q. I can do.

Thanks, Jinjing. I am trying to generalize your method since you mensioned 9781 is a prime. Then let's take a look at another equation: x^2 - 83*y^2 = 1. 83 is a prime.

Let y = 3^2, then (x+1)(x-1)= 83*3^4=83*81, (x+1)-(x-1) = (83*3^0 - 3^4)*3^0 = 2. Then we have x+1=83 and x-1=81, so (82, 9) is an intergral root. Notice that 3^0, that is the issue.

You method is more promising if you can amend a little bit. Even computer cannot solve number theory problems because of the storage capacity limitation. I suspect the following equations have no integral soltions either.

x^2 - 61*y^2 = 1

x^2 - 89*y^2 = 1

x^2 - 97*y^2 = 1

所有跟帖： 

• We only check |y^2-P|=2,If there is no solution,the Q has no who -jinjing- ♀ (298 bytes) () 01/26/2011 postreply 16:50:24

• (n^2+1)^2-(n^2+2)*n^2=1,(n^2-1)^2-(n^2-2)*n^2=1.If for any coeff -jinjing- ♀ (229 bytes) () 01/26/2011 postreply 19:08:09

• 回复：(n^2+1)^2-(n^2+2)*n^2=1,(n^2-1)^2-(n^2-2)*n^2=1.If for any co -calligraphy- ♂ (511 bytes) () 01/26/2011 postreply 20:47:19

• Your Ex,5should be 5^2m,3 ...3^2n...,We get |9781-Z^2|=1, -jinjing- ♀ (344 bytes) () 01/27/2011 postreply 09:35:02

• 回复：Your Ex,5should be 5^2m,3 ...3^2n...,We get |9781-Z^2|=1, -calligraphy- ♂ (425 bytes) () 01/27/2011 postreply 11:26:44

• 回复：回复：Your Ex,5should be 5^2m,3 ...3^2n...,We get |9781-Z^2|=1, -jinjing- ♀ (438 bytes) () 01/27/2011 postreply 17:19:24

• 回复：We only check |y^2-P|=2,If there is no solution,the Q has no -calligraphy- ♂ (162 bytes) () 01/27/2011 postreply 07:03:34

• 回复：We only check |y^2-P|=2,If there is no solution,the Q has no -calligraphy- ♂ (3010 bytes) () 01/27/2011 postreply 07:11:39

• 回复：We only check |y^2-P|=2,If there is no solution,the Q has no -calligraphy- ♂ (380 bytes) () 01/27/2011 postreply 07:48:25

• 回复：We only check |y^2-P|=2,If there is no solution,the Q has no -calligraphy- ♂ (201 bytes) () 01/27/2011 postreply 08:21:54