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Square products (Posted on 2018-02-27) Difficulty: 3 of 5
Find all integer solutions to (x3-1)(y3-1)=z2, with |x|<|y| and z>0.

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Some Thoughts computer exploration | Comment 1 of 3
The program doesn't list all the possibilities when x or y equals -1 as that would go on forever, but

 10    For tot = 0 To 10000
 20      For x0 = 0 To int((tot - 1) / 2)
 30        y0 = tot - x0
 40        x = -x0: y = -y0: GoSub *tryIt
 50        x = -x0: y = y0: GoSub *tryIt
 60        x = x0: y = -y0: GoSub *tryIt
 70        x = x0: y = y0: GoSub *tryIt
 90      Next
100    Next
110    
120    print "done"
130    
140    end
150    
160   *tryIt:
170      z2 = (x ^ 3 - 1) * (y ^ 3 - 1)
180      If z2 > 0 Then
190      :z = Int(Sqr(z2) + 0.5)
200      :If z * z = z2 Then
210        :print x , Str(y) ,Str(z) 
220      :End If
230      :End If
240   Return

finds 

(x,y,z) =

(0,-2,3)
(2,4,21)
(-1,-23,156)
(2,22,273)
(4,22,819)
(-6,-26,1953)
3,313,28236)
(-20,-362,616077)

The program at this point has gone through values where abs(x)+abs(y)<=896.

  Posted by Charlie on 2018-02-27 14:38:46
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