Integer Question

Experts,

Can you tell me the only integer N, where N-1 and N+1 are a square and a cube?


Thanks!
Marv1nAsked:
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d-glitchCommented:
26 -1 = 25 = 5*5
26+1 = 27 = 3*3*3
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d-glitchCommented:
No guarantee that is the only case though....
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phoffricCommented:
0 - 1 = -1 = (-1)*(-1)*(-1)
0 +1 = 1 = 1*1

Since you didn't say
N-1 and N+1 are a square and a cube, respectively
the above works non-respectively.
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dsackerContract ERP Admin/ConsultantCommented:
If N-1 must be the square and N+1 must be the cube, seems like 26 is the only number that fits the bill. If it can be either, you'll have a lot more numbers.

If you want to find numbers higher than that, set your maximum in this VBA snippet and run to  your heart's content:

Option Explicit
Public Sub TestThis()
    Dim ndx As Double
    Dim s As Double
    Dim c As Double

    For ndx = 1 To 999999999
        s = sqr(ndx - 1)
        c = (ndx + 1) ^ (1 / 3)
        If s = Round(s, 0) And c = Round(c, 0) Then
           Debug.Print ndx
        End If
        DoEvents
    Next ndx
End Sub

Open in new window

If you want to check either/or, this code will give many more numbers.

Option Explicit
Public Sub TestThis()
    Dim ndx As Double
    Dim s1 As Double
    Dim c1 As Double
    Dim s2 As Double
    Dim c2 As Double

    For ndx = 0 To 999999999
        If ndx > 0 Then
           s1 = sqr(ndx - 1)
        Else
           s1 = 2
        End If
        c1 = (ndx + 1) * (ndx + 1) * (ndx + 1)
        s2 = sqr(ndx + 1)
        c2 = (ndx - 1) * (ndx - 1) * (ndx - 1)
        If s1 = Round(s1, 0) And c1 = Round(c1, 0) Then
           Debug.Print ndx
        End If
        If s2 = Round(s2, 0) And c2 = Round(c2, 0) Then
           Debug.Print ndx
        End If
        DoEvents
    Next ndx
End Sub

Open in new window

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dsackerContract ERP Admin/ConsultantCommented:
Hope this wasn't a classroom homework assignment, else I'm telling your teacher. :)
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phoffricCommented:
>> else I'm telling your teacher
Heh, maybe it was. Not sure what grade though given the questions that Marv1n has been asking over the years. (You can see what questions he asks by clicking on his name and go to the Activity tab, and then the Questions tab.)
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Marv1nAuthor Commented:
Nah, someone asked me this in passing and you guys are way smarter than I ;)

I'm well out of school.
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TommySzalapskiCommented:
I'm curious how you would go about proving that 26 is the only one.
Essentially, you are saying that if sqrt(x^3 - 2) is an integer, then x = 3.
Hmm....
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dsackerContract ERP Admin/ConsultantCommented:
Read this thread through. The scripts above answered your question before you asked it.
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TommySzalapskiCommented:
The script would only prove it if it went from 1 to infinity, which it can't, you'd have issues with your double precision long before that.
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dsackerContract ERP Admin/ConsultantCommented:
Well DUH. Define the field type to allow larger values, and go far enough to satisfy that you are very unlikely to find a number in high exponential ranges where a distance of 2 will render a perfect square and cube. The script was for the thinking man. :)
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TommySzalapskiCommented:
If you pick an integer purely at random, the odds that it will be small enough to fit in the RAM of any computer in existence today is statistically 0.

So the script demonstrates that it holds true for roughly 0% of the integers. That may be enough for some people, but not for me.
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dsackerContract ERP Admin/ConsultantCommented:
Enjoy your personal world. All the best.
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phoffricCommented:
To deal with the OP question "the only integer", you need a mathematical proof that deals with all numbers, not just the first zillion numbers.

>> you are very unlikely to find a number in high exponential ranges where a distance of 2 will render a perfect square and cube.
    It may be that your intuition is correct. Then again, you may be wrong. And even if correct, you are only claiming "unlikely" as opposed to "impossible". It may be hard or even impossible to prove or disprove mathematically the OP conjecture.

If I had time, I would at least start considering some known number rules such as:
   the cube of integer can be written as the difference of two square or similarly,
for i=1..N sum(i^3) = (N+1)^2.

If the OP had included a comment "for the first quadrillion positive integers", then your extended computer solution would be adequate.
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