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 Expression Product Exercise (Posted on 2015-09-18)
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.)
 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 product2 13 644 4096.000000000025 23323703840.99996 6.37034642160154E+257 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

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

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