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.
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
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Posted by Charlie
on 2015-10-25 14:44:58 |
computer solution
