What Are Sums of Integer Powers Called in Number Theory?

[fukon]

G'day, geniuses!

I've been doing some experimental number theory and have found some interesting results, but I don't know how to find out if I'm simply rediscovering old ground. I need to find appropriate keywords / nomenclature for what I'm treating.

I am investigating sums of M terms, each term being an integer raised to the power N (N can be <M, =M, >M). Can anyone tell me what such an animal called, and where to look for the important findings?

Thanks in advance.

 

[Robert1986]

G'day, geniuses!

I've been doing some experimental number theory and have found some interesting results, but I don't know how to find out if I'm simply rediscovering old ground. I need to find appropriate keywords / nomenclature for what I'm treating.

I am investigating sums of M terms, each term being an integer raised to the power N (N can be <M, =M, >M). Can anyone tell me what such an animal called, and where to look for the important findings?

Thanks in advance.

What are the some of the results? This doesn't look familiar to me; at least nothing that I have seen. But that might be because I don't know in which context you are investigating them. Post some specific examples; I'll look at them, but I don't profess to be any sort of number theorist; sounds interesting, though.

 

[mathman]

Finite sums of fixed powers have been studied extensively. There is a classic book "Table of Integrals, Series, and Products" by Gradshteyn and Ryznik which includes these sums, with explicit results for N up to 7. The general result is always a polynomial in M of degree N+1.

 

[g_edgar]

$$\sum_{N=1}^M x^N$$
is called a "geometric series" is that what you mean?

 

[Robert1986]

I don't know for sure, but I assume (and I based my post on this assumption) that he has done something interesting with these sums. I'm thinking he has some specific results.

 

[mathman]

$$\sum_{N=1}^M x^N$$
is called a "geometric series" is that what you mean?

I think he means:

$$\sum_{k=1}^M k^N$$

 

[fukon]

Mathman's formulation is correct. I'd call it a finite power series of integers; does that sound right?

Anyway, I'm looking for specific properties and patterns in sums that exhibit these. And, yes, Robert1986: I have found some interesting results. I'd love to share them with you, but I want to make a bit more headway on my own, first.

I'm thinking about exploring a theoretical explanation for my findings, but I want to do a bit (well, probably more than a bit) of research before I risk making a fool of myself. I have found some useful book excerpts on Google Scholar, but I would prefer to get some whole books. If anybody would care to recommend a good number theory reference that covers this kind of thing I would be most grateful.

Nb. My mathematics training only goes as far as the requirements of a Canadian Physics degree (i.e. 3rd-year calculus, 2nd-year linear algebra), so I'm ignorant of much of the theory and jargon at level that I'm getting into.

Thanks, all!

 

[mathman]

The reference I gave (Gradshteyn and Ryznik) does have a general formula in terms of Bernoulli numbers as well as source references.

 

[fukon]

Thanks again, Mathman. I'll see if I can find it at the nearest university library.

BTW: I've been testing my hypothesis by running tests in a computer program (I'm calling this 'experimental number theory' until I'm advised of more appropriate jargon). I devised an algorithm that uses what I call 'virtual looping' to eliminate the need to keep increasing the nesting level of the loops every time I want to increment the number of terms in my series.

Nb. I'm not actually looking at the sum of powers from 0 to M, but rather the sum of M terms arbitrary chosen from the sequence 0..L, where L is bounded by computational constraints. I was getting away with L = 100 until I tried N (the exponent) = 9. Unfortunately, once N reached 9, my Delphi program went wonky; the int64 variable type was overwhelmed, and artificially negative numbers started turning up in the output. I'll have to do something tricky if I want to go further computationally. Rather than write my own code to handle, say, int128 numbers, is there an API out there I could use? (P.S. I realize this should be a new thread.)