N is a positive integer which is expressible as the sum of cubes of two positive integers.
Given that N is not divisible by 9, find the possible remainders when N is divided by 63.
All sums of two cubes a^3 + b^3 where a+b totaled no more than 10,000 were tested. Remainders, r, all were found with totals a+b summing no more than 23 (plus additional times at higher values, of course).
lowest
r N cubes roots
1 1072 343 729 7 9
2 2 1 1 1 1
7 133 8 125 2 5
8 9269 8 9261 2 21
16 16 8 8 2 2
19 1027 27 1000 3 10
20 1343 343 1000 7 10
26 152 27 125 3 5
28 28 1 27 1 3
29 344 1 343 1 7
34 2869 125 2744 5 14
35 35 8 27 2 3
37 730 1 729 1 9
43 2752 8 2744 2 14
44 737 8 729 2 9
47 2000 1000 1000 10 10
55 370 27 343 3 7
56 1001 1 1000 1 10
61 250 125 125 5 5
62 2771 27 2744 3 14
from
DefDbl A-Z
Dim crlf$, had(62, 2)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For tot = 2 To 10000
For a = 1 To tot / 2
b = tot - a
n = a * a * a + b * b * b
q = Int(n / 9)
r = n - 9 * q
If r > 0 Then
q = Int(n / 63)
r = n - 63 * q
If had(r, 0) = 0 Then
had(r, 0) = n: had(r, 1) = a: had(r, 2) = b
End If
End If
Next
Next
For i = 0 To 62
If had(i, 0) Then
Text1.Text = Text1.Text & mform(i, "##") & mform(had(i, 0), "####0") & " "
Text1.Text = Text1.Text & mform(had(i, 1) ^ 3, "####0") & mform(had(i, 2) ^ 3, "####0") & " "
Text1.Text = Text1.Text & mform(had(i, 1), "####0") & mform(had(i, 2), "####0") & crlf
End If
Next
Text1.Text = Text1.Text & crlf & crlf & " done"
End Sub
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
If a were not allowed to equal b the list would have the same set of remainders
1 1072 343 729 7 9
2 65 1 64 1 4
7 133 8 125 2 5
8 9269 8 9261 2 21
16 520 8 512 2 8
19 1027 27 1000 3 10
20 1343 343 1000 7 10
26 152 27 125 3 5
28 28 1 27 1 3
29 344 1 343 1 7
34 2869 125 2744 5 14
35 35 8 27 2 3
37 730 1 729 1 9
43 2752 8 2744 2 14
44 737 8 729 2 9
47 3197 1000 2197 10 13
55 370 27 343 3 7
56 1001 1 1000 1 10
61 5038 125 4913 5 17
62 2771 27 2744 3 14
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Posted by Charlie
on 2016-11-10 12:42:32 |
computer solution
