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

Home > Just Math
Sum Equality and Perfect Square (Posted on 2015-10-25) Difficulty: 3 of 5
Each of P, Q and R is a nonzero integer such that:
  • P+Q+R = P*Q + Q*R + R*P and,
  • P + Q + R is a perfect square.
Find the four smallest values of abs(P*Q*R)

*** abs(x) refers to the absolute value of x.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution Comment 2 of 2 |
In order of total abs(P)+abs(Q)+abs(R) up to that totalling 10,000:

abs(product)       P Q   R
           384    -4 8 12
         21504    -16 24 56
        120000    -30 50 80
        109824    -22 26 192
        428064    -42 56 182
       1077504    -64 122 138
       5040000    -100 140 360
       4472160    -88 110 462
      15205344    -154 264 374
       8163264    -102 122 656
      38385984    -208 338 546
      40903104    -214 362 528
      30474240    -160 192 992
      79236864    -238 306 1088
     153842304    -264 308 1892
     574080000    -520 920 1200
      21859200    -88 92 2700
     136819584    -216 236 2684
     814118304    -552 782 1886
      68465664    -144 152 3128
     696009600    -468 572 2600
     373699584    -328 368 3096
    1546406784    -676 936 2444
     622797504    -406 464 3306
    2610854784    -856 1436 2124
    2921673600    -900 1772 1832
    2577429504    -808 1136 2808
    3304387584    -904 1368 2672
    3707956224    -952 1512 2576
    4213730304    -1008 1736 2408
    2129857344    -592 666 5402
    9831972864    -1344 2432 3008
    4172186304    -814 962 5328
   12347744640    -1428 2312 3740
   14837592384    -1462 2096 4842
   17722609344    -1584 2402 4658
   23525262144    -1798 3258 4016
   14361600000    -1360 1760 6000
   24237031104    -1822 3536 3762
   13845395904    -1264 1538 7122
   
Since it's in order of total of the absolute values rather than product, we can't be fully sure (unless someone has a proof) that the lowest four products are included, but you can see that so far the size of the numbers has occasionally gone back one digit, but no farther.  Putting these into product order:

abs(product)       P Q   R
           384    -4 8 12
         21504    -16 24 56
        109824    -22 26 192
        120000    -30 50 80
        428064    -42 56 182
       1077504    -64 122 138
       4472160    -88 110 462
       5040000    -100 140 360
       8163264    -102 122 656
      15205344    -154 264 374
      21859200    -88 92 2700
      30474240    -160 192 992
      38385984    -208 338 546
      40903104    -214 362 528
      68465664    -144 152 3128
      79236864    -238 306 1088
     136819584    -216 236 2684
     153842304    -264 308 1892
     373699584    -328 368 3096
     574080000    -520 920 1200
     622797504    -406 464 3306
     696009600    -468 572 2600
     814118304    -552 782 1886
    1546406784    -676 936 2444
    2129857344    -592 666 5402
    2577429504    -808 1136 2808
    2610854784    -856 1436 2124
    2921673600    -900 1772 1832
    3304387584    -904 1368 2672
    3707956224    -952 1512 2576
    4172186304    -814 962 5328
    4213730304    -1008 1736 2408
    9831972864    -1344 2432 3008
   12347744640    -1428 2312 3740
   13845395904    -1264 1538 7122
   14361600000    -1360 1760 6000
   14837592384    -1462 2096 4842
   17722609344    -1584 2402 4658
   23525262144    -1798 3258 4016
   24237031104    -1822 3536 3762

The lowest four products are    384, 21504, 109824 and 120000.

DefDbl A-Z
Dim crlf$

 Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
 End Function



Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 

 
 For tot = 3 To 10000
   DoEvents
   For p0 = 1 To tot / 3
     For q0 = p0 To (tot - p0) / 2
       r0 = tot - p0 - q0
       For p = -p0 To p0 Step 2 * p0
       For q = -q0 To q0 Step 2 * q0
       For r = -r0 To r0 Step 2 * r0
         If p + q + r = p * q + q * r + r * p Then
           sq = p + q + r
           If sq >= 0 Then
            sr = Int(Sqr(sq) + 0.5)
            If sr * sr = sq Then
              Text1.Text = Text1.Text & mform(Abs(p * q * r), "#############0") & "    " & p & Str(q) & Str(r) & crlf
            End If
           End If
         End If
       Next
       Next
       Next
     Next
   Next
 Next


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



  Posted by Charlie on 2015-10-25 14:44:58
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 (3)
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