+140 A lattice point in the \(x,y\) -plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation \(x^2-y^2=47\)?
+118667 mmm
\(x^2-y^2=47\\ x^2=47+y^2\\ 47+y^2=x^2 \)
so 47+ some square number = another square number
here are the smallest square numbers. I will stop when the difference between the biggest and the one just below it is more than 47
1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576
576-529=47 so I found one (actually two). 24^2-23^2=47
(24,23)
(-24,-23)
No bigger squares can work
Maybe there are some others in there, you can look, but I doubt it.
+118667 Mmm
1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576
1+47=47
4+47+51
9+47=56
16+47=63
25+47=72
36+47=83
49+47=96
64+47=111
81+47=128
100+47=147
121+47=168
144+47=191
169+47=216
196+47=243
225+47=272
256+47=303
289+47=336
324+47=371
361+47=408
400+47=447
441+47=488
484+47=531
529+47=576 I already had that one.
I do not think there are any others.
