All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Pythagorean H/L ratios (Posted on 2017-07-08) Difficulty: 3 of 5
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.

No Solution Yet Submitted by Jer    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution 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    0.00352
10065 11752 15473    1.53730750124193    0.00269
10472 12225 16097    1.53714667685256    0.00285
10980 12859 16909    1.53998178506375    0.00002
11500 13509 17741    1.54269565217391    0.00270
12127 14136 18625    1.53582914158489    0.00417
12903 15104 19865    1.53956444237774    0.00044
13703 16104 21145    1.54309275341166    0.00309
14664 17177 22585    1.5401663938898     0.00017
15225 17792 23417    1.53806239737274    0.00194
15264 17927 23545    1.54251834381551    0.00252
16985 19992 26233    1.54448042390344    0.00448
17556 20467 26965    1.53594212804739    0.00406
18212 21285 28013    1.53816165165825    0.00184
18675 21868 28757    1.53986613119143    0.00013
18880 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$

Private Sub Form_Load()
 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
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information