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Counting ARATs (Posted on 2016-09-07) Difficulty: 3 of 5
Let (a, b, c) denote a triplet of distinct integers in an ascending order.
If a2+ b2=c2+1
or if
a2+ b2=c2- 1
we will call such a triplet an ARAT (since it represents an almost-right-angle triangle) .
(4,8,9) is such a triplet.

How many ARATs are there, provided c<1,000,000?

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts Corrected version for c<100,000 | Comment 3 of 4 |
The revised version shows 311,395 triplets with c less than 100,000. The first 100 (in order of increasing a+b) are shown:

4 7 8     
4 8 9     *
8 9 12     
7 11 13     
6 17 18     
6 18 19     *
11 13 17     
10 15 18     
9 19 21     
14 17 22     
13 19 23     
17 21 27     
8 31 32     
16 23 28     
8 32 33     *
11 29 31     
15 26 30     
14 31 34     
20 25 32     
19 27 33     
18 30 35     *
17 34 38     
23 29 37     
22 31 38     
13 41 43     
16 41 44     
10 49 50     
26 33 42     
10 50 51     *
25 35 43     
19 43 47     
23 41 47     
31 34 46     
29 37 47     
28 39 48     
22 46 51     *
15 55 57     
34 38 51     *
32 41 52     
21 53 57     
25 49 55     
31 43 53     
24 55 60     
35 45 57     
34 47 58     
12 71 72     
12 72 73     *
20 65 68     
23 64 68     
31 56 64     
38 49 62     
17 71 73     
37 51 63     
26 65 70     
41 53 67     
40 55 68     
29 67 73     
33 64 72     
49 50 70     
22 79 82     
25 76 80     
44 57 72     
43 59 73     
28 76 81     *
41 64 76     
51 55 75     
19 89 91     
31 77 83     
47 61 77     
46 63 78     
39 71 81     
14 97 98     
14 98 99     *
44 68 81     *
34 79 86     
43 71 83     
50 65 82     
27 89 93     
49 67 83     
41 79 89     
53 69 87     
52 71 88     
36 89 96     
56 73 92     
21 109 111     
39 91 99     
55 75 93     
29 103 107     
32 100 105     *
47 86 98     
35 99 105     
59 77 97     
26 111 114     
58 79 98     
69 71 99     
65 76 100     
41 101 109     
16 127 128     
49 94 106     
62 81 102  

Asterisk indicates a^2 + b^2 < c^2 (by 1 of course).

Intermediate totals are shown for increments of 5000 total a+b:

 8013 5000     17795 10000     28249 15000     39092 20000     50297 25000     61813 30000     73381 35000     85245 40000     97192 45000     109350 50000     121647 55000     134021 60000     146514 65000     159070 70000     171773 75000     184488 80000     197281 85000     210189 90000     223189 95000     236269 100000     249196 105000     261363 110000     272695 115000     283099 120000     292442 125000     300337 130000     306683 135000     311022 140000    

311395 done

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 
 For tot = 3 To 141422
   For a = 1 To tot / 2
    DoEvents
     b = tot - a
     If b > a Then
        a2 = a * a: b2 = b * b
        c2 = a2 + b2
        sr = Int(Sqr(c2) + 0.5)
        If Abs(sr * sr - c2) = 1 And sr < 100000 And sr > b Then
          ct = ct + 1
          If ct < 101 Then
            Text1.Text = Text1.Text & a & Str(b) & Str(sr) & "     "
            If (sr * sr - c2) > 0 Then
             Text1.Text = Text1.Text & "*" & crlf
            Else
             Text1.Text = Text1.Text & crlf
            End If
          End If
        End If
     End If
   Next
   If tot Mod 5000 = 0 Then Text1.Text = Text1.Text & Str(ct) & Str(tot) & "    "
 Next
 
 
 
 Text1.Text = Text1.Text & crlf & crlf & ct & " done"
  
End Sub


  Posted by Charlie on 2016-09-07 12:57:02
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