 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Square Egyptians (Posted on 2017-06-28) Express 1/2 as the sum of Square Egyptian Numbers with distinct denominators under 362.

I.e. reciprocals of perfect squares.

 No Solution Yet Submitted by Jer No Rating Comments: ( Back to comment list | You must be logged in to post comments.) computer solution Comment 1 of 1
1/2 = 1/4 + 1/9 + 1/16 + 1/25 + 1/49 + 1/144 + 1/225 + 1/400 + 1/784 + 1/1225

The denominators are the squares of:

2 3 4 5 7 12 15 20 28 35

as found by the below VB program and verified by using exact rational fractions in UBASIC.

The program  produces a total that prints out as exactly .5 but did not register as equal to .5 within the program, due to rounding, and thus necessitated the difference to be below the small threshhold of absolute value difference shown in the program, rather than testing if = 0.5.

The first portion of the program determined how far back from 1/36^2 would be necessary and found that the first term must be 1/2^2, otherwise the total would be impossible. It was only after this was run that the rest of the program was added, to find the actual result.

DefDbl A-Z
Dim crlf\$, h(36), tot

Form1.Visible = True

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

For rt = 36 To 1 Step -1
ptot = tot
tot = tot + 1 / (rt * rt)
If tot >= 0.5 Then Text1.Text = Text1.Text & rt & Str(tot) & Str(ptot) & crlf & crlf: Exit For
Next

h(1) = 4: tot = 1 / 4

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

End Sub

DoEvents
st = Int(Sqr(h(wh - 1)) + 0.5) + 1
For i = st To 36
h(wh) = i * i
savetot = tot
tot = tot + 1 / (i * i)
If Abs(tot - 0.5) < 0.000000000001 Then
Text1.Text = Text1.Text & tot & crlf & "   "
For j = 1 To wh
Text1.Text = Text1.Text & " + 1/" & h(j)
Next
Text1.Text = Text1.Text & crlf
For j = 1 To wh
Text1.Text = Text1.Text & Str(Sqr(h(j)))
Next
Text1.Text = Text1.Text & crlf

End If
If i < 36 And tot < 0.5001 Then
End If
tot = savetot
Next
End Sub

2 .617538519845469 .367538519845469

0.5
+ 1/4 + 1/9 + 1/16 + 1/25 + 1/49 + 1/144 + 1/225 + 1/400 + 1/784 + 1/1225
2 3 4 5 7 12 15 20 28 35

 Posted by Charlie on 2017-06-28 13:59:04 Please log in:

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