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 Pythagorean H/L ratios (Posted on 2017-07-08)
Find the smallest pythagorean triple a<b<c where c/a is within 0.005 of 1.54 and then do the same for c/b.

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 computer solution | Comment 1 of 2
Quantity b is the length of the longer leg of the right triangle with c as the hypotenuse.  The ratio of c to b can never exceed sqrt(2), so the second part is impossible.

For part one, the answer is (133, 156, 205), the first such on the following table showing all cases where c is under 30,000 and GCD(a,b,c)=1.

`    a    b     c          c/a           off target  133   156   205    1.54135338345865    0.00135  540   629   829    1.53518518518519    0.00481 1564  1827  2405    1.53772378516624    0.00228 1764  2077  2725    1.54478458049887    0.00478 1971  2300  3029    1.53678335870117    0.00322 3120  3649  4801    1.53878205128205    0.00122 3400  3999  5249    1.54382352941176    0.00382 4551  5320  7001    1.53834322127005    0.00166 5031  5920  7769    1.54422580003975    0.00423 5208  6095  8017    1.53936251920123    0.00064 5568  6545  8593    1.54328304597701    0.00328 5969  6960  9169    1.53610319986597    0.00390 7400  8631 11369    1.53635135135135    0.00365 7828  9165 12053    1.53972917731221    0.00027 8195  9588 12613    1.53910921293472    0.00089 8268  9715 12757    1.5429366231253     0.00294 8835 10388 13637    1.54352009054895    0.0035210065 11752 15473    1.53730750124193    0.0026910472 12225 16097    1.53714667685256    0.0028510980 12859 16909    1.53998178506375    0.0000211500 13509 17741    1.54269565217391    0.0027012127 14136 18625    1.53582914158489    0.0041712903 15104 19865    1.53956444237774    0.0004413703 16104 21145    1.54309275341166    0.0030914664 17177 22585    1.5401663938898     0.0001715225 17792 23417    1.53806239737274    0.0019415264 17927 23545    1.54251834381551    0.0025216985 19992 26233    1.54448042390344    0.0044817556 20467 26965    1.53594212804739    0.0040618212 21285 28013    1.53816165165825    0.0018418675 21868 28757    1.53986613119143    0.0001318880 22119 29081    1.54030720338983    0.00031`

The output of the following was massaged for the above to line up the columns.

DefDbl A-Z
Dim crlf\$

Form1.Visible = True

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

For c = 1 To 30000
val0 = c / 1.54
For a = Int(val0) To 1 Step -1
DoEvents
If c / a > 1.545 Then Exit For
b2 = c * c - a * a
b = Int(Sqr(b2) + 0.5)
If b * b = b2 And gcd(gcd(a, b), c) = 1 Then
Text1.Text = Text1.Text & a & Str(b) & Str(c) & "    " & c / a & "    " & Format\$(Abs(c / a - 1.54), "0.00000") & crlf
End If
Next a
For a = Int(val0) + 1 To c - 1
DoEvents
If c / a < 1.535 Then Exit For
b2 = c * c - a * a
b = Int(Sqr(b2) + 0.5)
If b * b = b2 And gcd(gcd(a, b), c) = 1 Then
Text1.Text = Text1.Text & a & Str(b) & Str(c) & "    " & c / a & "    " & Format\$(Abs(c / a - 1.54), "0.00000") & crlf
End If
Next a
Next c

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

End Sub

Function gcd(a, b)
x = a: y = b
Do
q = Int(x / y)
z = x - q * y
x = y: y = z
Loop Until z = 0
gcd = x
End Function

 Posted by Charlie on 2017-07-08 11:49:48

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