 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Arithmetic Triplet Ascertainment (Posted on 2016-01-03) F(n) denotes the largest prime divisor of a positive integer n.

Determine all possible triplets (A,B,C) of strictly increasing positive integers such that:
• A, B and C are in arithmetic sequence, and:
• F(A*B*C) ≤ 3

 No Solution Yet Submitted by K Sengupta No Rating Comments: ( Back to comment list | You must be logged in to post comments.) Heuristic solution Comment 1 of 1
These three sets form the basis of an infinite number of sets:

(1, 2, 3)
(2, 3, 4)
(2, 9, 16)

multiply any one of these three sets by any number that's a product of powers of 2 and 3, and you get another set.

Before the GCD test in the below program was added, many more were found. The GCD test showed all are based on the above three, having tested all cases where a+c <= 30,000.

DefDbl A-Z
Dim crlf\$

Form1.Visible = True

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

For tot = 4 To 30000
For a = 1 To (tot - 1) / 2
DoEvents
If test(a) Then
c = tot - a
b = (a + c) / 2
If b = Int(b) Then
If gcd(a, b) = 1 Then
If test(b) Then
If test(c) Then
Text1.Text = Text1.Text & a & Str(b) & Str(c) & crlf
End If
End If
End If
End If
End If
Next
Next

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

End Sub

Function test(x)
t = x
While t Mod 2 = 0
t = t / 2
Wend
While t Mod 3 = 0
t = t / 3
Wend
If t = 1 Then test = 1 Else test = 0
End Function

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 2016-01-03 10:57:44 Please log in:

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