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

Home > Just Math
Expression Product Exercise (Posted on 2015-09-18) Difficulty: 3 of 5
Prove that when all expessions of the form:
+/- √1 +/- √2 +/- ....... +/- √100 are multiplied together, the result is an integer.
As an example, multiplying all expressions of the form: +/- √1 +/- √2 is equivalent to finding the result of this product:
(√1 +√2)( √1 - √2)( - √1 + √2)( - √1 - √2)

**** Extra Challenge: Solve this puzzle without the aid of a computer program.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts computer exploration -- far from a proof | Comment 1 of 5
If n is how high many terms is used per factor (largest number whose square root is taken):

n resulting product
2 1
3 64
4 4096.00000000002
5 23323703840.9999
6 6.37034642160154E+25
7 3.16699666163359E+59

For n>5 we don't really have an indication whether the function results in an integer or not, but below 6 it looks as if the non-integral part is a result of rounding error. For n>4 they are not necessarily powers of 2; the numbers for n=5 through 7 definitely are not.

For n=5 the factors are (just as a check):

-4.14626436994197
-.682162754804218
-1.31783724519578
 2.14626436994197
-2.14626436994197
 1.31783724519578
 .682162754804218
 4.14626436994197
 
DefDbl A-Z
Dim crlf$, n


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 For n = 2 To 7
 prod = 1
 ReDim term(n)
 For i = 1 To n
  term(i) = Sqr(i)
 Next
 For f = 0 To Int(2 ^ n - 0.5)
   tot = 0
   bs$ = "": f1 = f
   Do
     d = f1 Mod 2: f1 = f1 \ 2
     bs = LTrim(Str(d)) + bs
   Loop Until f1 = 0
   bs = Right(String(n, "0") + bs, n)
   For i = 1 To Len(bs)
     If Mid(bs, i, 1) = "0" Then
       tot = tot - term(i)
     Else
       tot = tot + term(i)
     End If
   Next
   prod = prod * tot
   If n = 3 Then
     Text1.Text = Text1.Text & Str(tot) & crlf
   End If
 Next
 Text1.Text = Text1.Text & n & Str(prod) & crlf
 Next n

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

I don't really have time today to redo the project in UBASIC to get higher precision, to assure the integral nature for a couple more levels of n. But even if done in UBASIC, we couldn't get even close to 100, due to time and precision limitations. 

  Posted by Charlie on 2015-09-18 12:37:49
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 (1)
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