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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 \(x^2-y^2=47\)?

Aug 14, 2021

#2
+1

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.

Aug 15, 2021
#3
+1

That was my originaly answer and it's wrong apparently

sherwyo  Aug 15, 2021
#4
+1

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

I do not think there are any others.

Melody  Aug 15, 2021
#5
+1

I get two lattice points: Aug 15, 2021
#6
+3

Hi Alan

(24,23)

(-24,-23)

(24,-23)

I guess  (-24,23)  can also be added.

So now we have 4 points

Melody  Aug 15, 2021
#7
+1

True.  I should have noted that the factors could be -1 and -47 as well as 1 and 47.

Alan  Aug 15, 2021