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Near Fermat Nuance (Posted on 2016-08-30) Difficulty: 3 of 5
Consider the equation: X3 + Y3 = Z3, where

(i) Each of X and Y is a positive integer and

(ii) Z is a real number such that Z3 is a positive integer, and:

(ii) The four digits immediately following the decimal point in the base ten expansion of Z are 2, 0, 1 and 6 (in this order)

Find the three smallest values of X+Y

No Solution Yet Submitted by K Sengupta    
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Solution computer solution Comment 2 of 2 |
The three smallest values of x+y are 191, 278 and 333 based on this table, where x is chosen as the smaller of x and y.

   x  y     x+y           z^3           z
  64 127    191         2310527      132.201692732851
 115 163    278         5851622      180.201646447575
 159 174    333         9287703      210.20164305405
  39 313    352        30723616      313.201699145934
 163 212    375        13858875      240.201653435429
 171 239    410        18652130      265.201602848263
  45 388    433        58502197      388.201663420858
  46 401    447        64578537      401.201671764029
 100 388    488        59411072      390.201675223557
 165 360    525        51148125      371.201656800751
 179 347    526        47517262      362.201687513729
  52 482    534       112120776      482.201656866414
  53 496    549       122172813      496.201635252072
 162 440    602        89435528      447.201610012093
 101 534    635       153303605      535.201665028562
 283 367    650        72096050      416.201675049523
 303 351    654        71061678      414.201645100923
 104 558    662       174865976      559.201642985397
 129 569    698       186366698      571.201628375468
 132 589    721       206636437      591.201646524602
 186 585    771       206636481      591.201688486983
  67 705    772       350703388      705.201651150452
 246 584    830       214063640      598.201689591674
  71 769    840       455114520      769.201691321344
 281 611    892       250287172      630.20164201339
  75 835    910       582604750      835.201643724776
 274 664    938       313325768      679.201606218538
 467 496    963       223871499      607.201639804036
 300 675    975       334546875      694.201678105739
 
 
DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For tot = 2 To 1000
   For x = 1 To tot / 2
     DoEvents
     y = tot - x
     z3 = x * x * x + y * y * y
     If z3 < 100000000000# Then
       cr$ = Str(z3 ^ (1 / 3))
       ix = InStr(cr, ".")
       If Mid(cr, ix + 1, 4) = "2016" Then
          Text1.Text = Text1.Text & mform(x, "####") & mform(y, "####") & "   " & mform(tot, "####") & "      " & mform(z3, "##########") & "     " & cr & crlf
       End If
     End If
   Next
 Next
 
 
 Text1.Text = Text1.Text & crlf & ct & " done"
  
End Sub


Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function
 

  Posted by Charlie on 2016-08-30 15:03:22
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