go to the web address below

Bertie

I don't have a complete solution either but this is what I have so far.

\(\displaystyle f(n) = 9^{n+1}-8n-9\),   

so    

\(\displaystyle f(n+1) = 9^{n+2}-8(n+1)-9\) ,

in which case

\(\displaystyle \frac{f(n+1)}{f(n)}=\frac{9-\{8(n+1)+9\}/9^{n+1}}{1-\{8n+9\}/9^{n+1}}\),

having divided top and bottom by \(\displaystyle 9^{(n+1)}\).

That means that for increasing n, \(\displaystyle f(n+1) \approx 9f(n) \text{ or } f(n+1) \approx 2.08^{3}f(n)\) ,

and that implies that for each increase of 1 in n, m will increase by 3.

That effect seems to kick in early,

f(1) = 64 = 2^6, so m = 6

f(2) = 704 = 2^9 + 192, so m = 9

f(3) = 6528 = 2^12 + 2432 so m = 12

f(4) = 59008 = 2^15 + 26420 so m = 15

That in turn suggests the rule

m = 3n+3.

I've not yet convinced myself whether or not that this is true for all n.

-Bertie

3^2n +2 = k

3^2n = k-2

log_3(k-2)=2n

1/2(log_3(k-2)) = n

k is whatever the expression is equal to initally. you didn't say it was equal to anything at first

thats not even the question

i told you to go the link

Hi Aditya,

The address that you have provided is not correct. :(

Did you check it for yourself?

Anyway I am starting to feel sorry for you - you must be getting very frustrated by now.

[Aditya has been PMing me with this question for the better part of a day. ]

Here is your question

People can answer it here I guess.

if n belongs to natural numbers then the highest integer m such that 2^m divides (3^2n)9-8n-9

if n belongs to natural numbers then the highest integer m such that

2^m        divides     3^(2n+2)  -8n -9

I have looked at this question and have not made any immediate progress.

I am too busy to look properly now.

I will however draw attention to your question, so that other mathematicians will pay attention to it.

-----------------------------------------------------------------------

I have called for more help here:

http://web2.0calc.com/questions/repeated-difficult-question-please-help-is-you-can

THE STATEMENT IS FALSE. NO SOLUTIONS EXIST!.

Your statement may be true but it is valueless since it is comes with no proof or supporting argument!

I think that guest #4 is correct

http://www.wolframalpha.com/input/?i=%283%5E%282n%2B2%29++-8n+-9%29%2F2%5Ek+%3Dg+

I would still be interested in hearing a supporting argument. :)

Hallo aditya@calc.com,

sorry i have no closed solution. Here my way so far.

\(\small{ \begin{array}{rcl} 3^{2n+2} -8n -9 &=& 3^{2n}\cdot 3^2-8n-9 \\ &=& 3^{2n}\cdot 9-8n-9 \\ &=& (3^2)^n\cdot 9-8n-9 \\ &=& 9^n\cdot 9-8n-9 \\ &=& 9^{n+1}-8n-9 \\ &=& (1+8)^{n+1}-8n-9 \\ &=& -8n-9 + (1+8)^{n+1}\\ &=& -8n-9 + \underbrace{\binom{n+1}{0}+ \binom{n+1}{1}\cdot 8}_{=8n+9} + \binom{n+1}{2}\cdot 8^2 + \binom{n+1}{3}\cdot 8^3 + \binom{n+1}{4}\cdot 8^4 +\cdots \\ && + \binom{n+1}{n-1}\cdot 8^{n-1} + \binom{n+1}{n}\cdot 8^{n} + \binom{n+1}{n+1}\cdot 8^{n+1} \\\\ &=& \binom{n+1}{2}\cdot 8^2 + \binom{n+1}{3}\cdot 8^3 + \binom{n+1}{4}\cdot 8^4 +\cdots + \binom{n+1}{n-1}\cdot 8^{n-1} + \binom{n+1}{n}\cdot 8^{n} + \binom{n+1}{n+1}\cdot 8^{n+1} \\\\ &=& \binom{n+1}{2}\cdot 2^6 + \binom{n+1}{3}\cdot 2^9 + \binom{n+1}{4}\cdot 2^{12} +\cdots + \binom{n+1}{n-1}\cdot 2^{3n-3} + \binom{n+1}{n}\cdot 2^{3n} + \binom{n+1}{n+1}\cdot 2^{3n+3} \\ \end{array} }\)

It is not easy to factorize the binoms \(\binom{n+1}{2},~\binom{n+1}{3},~\binom{n+1}{3}\cdots\)  to get the exponent of the prim number 2 for every n

but m mimimum is 6, because we have \(2^6\)

It seems that (no proof !!!!)

  \(\begin{array}{rcl} \binom{n+1}{2}\cdot 2^6 &=& 2^m\cdot p \\\\ (n+1)\cdot n \cdot 2^5 &=& 2^m\cdot p \\ \end{array}\)

Example:

\(\begin{array}{|r|c|l|} \hline n & m & p \\ \hline 1 & 6 & 1 \\ 2 & 6 & 3 \\ \hline 3 & 7 & 3 \\ 4 & 7 & 5 \\ \hline 5 & 6 & 15 \\ 6 & 6 & 21 \\ \hline 7 & 8 & 7 \\ 8 & 8 & 9 \\ \hline 9 & 6 & 45 \\ 10 & 6 & 55 \\ \hline 11 & 7 & 33 \\ 12 & 7 & 39 \\ \hline 13 & 6 & 91 \\ 14 & 6 & 105 \\ \hline 15 & 9 & 15 \\ 16 & 9 & 17 \\ \hline 17 & 6 & 153 \\ 18 & 6 & 171 \\ \hline \cdots \\ \hline \end{array}\)

Here's my attempt (basically the same as heureka's):

I tried to upload this from my phone yesterday (I was away from my computer), but with no success!

OK., so that doesn't work.

Back to the drawing board.

-Bertie

Thanks Heureka, Alan and Bertie,

You are all terrific.        

I will send a message to aditya to make sure she sees your answers :)

Bertie

Updating my earlier posts #9 and #10, the sequence for m is

6,9,12,15, (as  in #9)

then 19 (an increase of 4 which caused me to write #10 because I thought #9 was wrong),

but it then continues

22,25,28,31,34, (steps of 3)

38 (step of 4)

41,44,47,50,53, (steps of 3)

57 (step of 4)

60,63,66,69,72, (steps of 3)

76 (step of 4).

i.e. five steps of 3 then one step of 4.

Whether or not this pattern continues indefinitely I don't know.

-Bertie

Thanks for the update Bertie :)