Powers of integers and factorials

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I would like some direction on studying powers of integers and if they are in any way related to factorials. I was studying the sequence of cubics 1, 8, 27, 64, 125 and so. After a certain number of rounds of a basic rule I choose to apply to this sequence, I arrived at a new sequence where one particular integer (not 0 nor 1) was repeated. I tried a sequence of integers raised to fourth power and found that my process brought about similar results. Depending on the exponent ( natural number ) used on the integers that I write out, I can now predict how many rounds it will take to get to the repeated integer and also predict that the repeated integer is a certain factorial. I am not referring to 0! nor 1! Does this discovery for me seem important or useful to any branch of math? I am new at asking questions here; I enjoy patterns with numbers and am trying to write conjectures or maybe a theorem or two from my discoveries. thanks for the help .
 

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  • #2
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I would like some direction on studying powers of integers and if they are in any way related to factorials. I was studying the sequence of cubics 1, 8, 27, 64, 125 and so. After a certain number of rounds of a basic rule I choose to apply to this sequence, I arrived at a new sequence where one particular integer (not 0 nor 1) was repeated. I tried a sequence of integers raised to fourth power and found that my process brought about similar results. Depending on the exponent ( natural number ) used on the integers that I write out, I can now predict how many rounds it will take to get to the repeated integer and also predict that the repeated integer is a certain factorial. I am not referring to 0! nor 1! Does this discovery for me seem important or useful to any branch of math? I am new at asking questions here; I enjoy patterns with numbers and am trying to write conjectures or maybe a theorem or two from my discoveries. thanks for the help .

You are going to have to tell us what rule you used.
 
  • #3
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For whatever its worth, without any reference to factorials, the digital root of n^3 is 1, 8, 9 repeating, while the digital root of the sum of cubes (Triangular Numbers Squared) is 1,9,9 repeating. As opposed, for instance, to the digital root of n^2, which is 1,4,9,7,7,9,4,1,9 repeating.

Curious what your "rule" is.
 
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  • #4
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Why would the rule be important in order to give me some direction? It is a simple rule applied to all terms of the sequence. Is there a known relationship between integers raised to a specific power and a factorial?
 
  • #5
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Why would the rule be important in order to give me some direction?
What's 31 + 31, sjohnsey?

Might the answer to that question not depend on what base was our mutual reference point?

Let me be the first to assure you: whatever you are or are not "on to" is probably quite trivial, even if correct.
 
  • #6
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Is there a known relationship between integers raised to a specific power and a factorial?
Sure, the nth series of differences of a power series ( a^n where "a" takes on the values 0,1,2,3...) is n! I gave a long proof of this fact to my High School teacher in 1962 and she told me that there was a simpler proof, but she couldn't recall what it was.
 
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  • #7
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Thanks for that help. I am in a bit of rush right now but I will look into that more. I am not really working with a series, just the sequence of integers. For example do you know how to relate 1^3, 2^3, 3^3, 4^3 and so on to 3! ?
 
  • #8
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Thanks for that help. I am in a bit of rush right now but I will look into that more. I am not really working with a series, just the sequence of integers. For example do you know how to relate 1^3, 2^3, 3^3, 4^3 and so on to 3! ?
It is the same thing when I said series I meant sequence(s) also. So now that I gave you a rule to get n!, what is your rule? or is it the same?
 
  • #10
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Thanks very much Dodo. That is very close to what I was looking for. I will study what K. W. has to offer!
 

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