positive

Primary tabs

Defines:
negative
Synonym:
greater than zero
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

Mathematics Subject Classification

proof

prove that

sum(a:1->n)a^b/sum(a:1->n)a

is an integer (b is odd)

induction

Re: proof

yeah yeah i was fast to answer, too fast actually,

but it feels like a double induction, on b and on n.

Re: proof

@ Tsurbaba

Can you please give the complete proof?

Re: proof

hi, actually i spoke too fast, i am quite lazy when coming down to manipulating equations, for me mathematics is a joyful art rather than an endless challenge.

, anyway to make a long story short, i gave this problem
to my father , who is a gifted equation-tamer (like a lion tamer, but more hard), so here is the solution , hope this catches you before
your assignment is due (how i hated this endless assignments..)

I just solved the problem you gave me, not by induction: Newton's binom:

N = sum(i=1:n) i^m m odd.
In reverse order: N = sum(i=1:n) (n+1-i)^m
Summing: 2N = sum(i=1:n) [i^m + (n+1-i)^m]

With Newton: (n+1-i)^m = (n+1)^m +... +(-i)^m

m is odd so (-i)^m cancels i^m. All other terms are multiple of n+1:

2N = K(n + 1)

Easy to show that 2N is also a multiple of n:

2N = 2sum(i=1:n-1) i^m + 2n^m

Done.

Re: proof

hi Tsurbaba

Thanks for your proof. But i think we need to show that
N is a multiple of n and n+1 simultaneoulsy

because we need to show that N is divisible by n(n+1)

for e.g. x is divisible by 4*3 means we need to show that x is divisible by 12 and not that x is divisible by 4 and x is divisible by 3(because 12 is divisible by 4 and 12 is divisible by 6 does not mean that 12 is divisible by 24)

please correct me if I am wrong

What do you say?

Re: proof

please ignore my above post. I got your point. Thanks a lot for helping me. I need another help from you. Whats the units digit in the expansion of

(15+sqrt220)^19 + (15+sqrt220)^82

stretchy letters

The variables in this entry look all stretchy, I don't know why.

Also, I'm wondering about the thebibliography thing. After "begin{thebibiliography}" there is a "{1}" or a "{5}" or "{2}." I've never seen "{3}" or "{8}" for some reason. What do these numbers mean?

Re: stretchy letters

In the HTML mode they seem now to be such. I think it is a system malfunction, which will soon be over.

Re: stretchy letters

If Mathnerd sees it, you see it and I see it, maybe a rerender needs to be done.

Re: stretchy letters

knodeltheory writes:

> maybe a rerender needs to be done.

Done.

For future reference, the URL for rerendering is
http://planetmath.org/?op=rerender&from=objects&id=xxxx
with xxxx replaced by the object ID number.