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 No software allowed ! (Posted on 2017-05-07)
Find positive integers a, b, and c, all different, such that
a^3 + b^3 = c^4.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 re(2): Solution | Comment 3 of 6 |

The sets that meet the criterion that all of a, b and c be different are marked with an * below.  The other sets satisfy the equation but contain at least one equality between number pairs.

2 2 2
9 18 9
32 32 16
28 84 28
70 105 35 *
162 162 54
65 260 65
144 288 72 *
273 364 91 *
126 630 126
266 665 133 *
512 512 128
469 603 134 *
456 760 152 *
455 1001 182 *
217 1302 217
793 854 183 *
756 945 189 *
448 1344 224 *
737 1072 201 *
730 1241 219 *
729 1458 243 *
417 1807 278 *
728 1729 273 *
1250 1250 250
344 2408 344
721 2060 309 *
1120 1680 280 *
702 2457 351 *
211 3165 422 *
665 2926 399 *
1110 2590 370 *
1705 2046 341 *
604 3473 453 *
1628 2849 407 *
513 4104 513
1136 3905 497 *
2592 2592 432
1040 4160 520 *
386 4825 579 *
2340 3276 468 *
217 5642 651 *
1617 4312 539 *
2304 4608 576 *
3354 3913 559 *
730 6570 730
1474 6633 737 *
3185 5096 637 *
3666 5170 658 *
2268 6804 756 *
4802 4802 686
4368 5824 728 *
3172 7137 793 *
1001 10010 1001
4270 7686 854 *
2016 10080 1008 *
5985 6840 855 *
3081 10270 1027 *
5670 8505 945 *
4256 10640 1064 *
1332 14652 1332
8192 8192 1024
7770 8806 1036 *
5625 11250 1125 *
7504 9648 1072 *
2678 14729 1339 *
7358 10754 1132 *
4074 14938 1358 *
7296 12160 1216 *
...

manually terminated

DefDbl A-Z
Dim crlf\$

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For tot = 3 To 9999999
For a = 1 To tot / 2
a3 = a * a * a
b = tot - a
b3 = b * b * b
c4 = a3 + b3
c = Int(Sqr(Sqr(c4)))
If c * c * c * c = c4 Then
Text1.Text = Text1.Text & a & Str(b) & Str(c)
If b > a And c <> b And c <> a Then Text1.Text = Text1.Text & " *"
Text1.Text = Text1.Text & crlf
End If
DoEvents
Next
Next tot

Text1.Text = Text1.Text & crlf & " done"

End Sub

 Posted by Charlie on 2017-05-07 16:26:51

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