The 4^n Polynomial conjecture

[Marc LeBrun]

 >=Cino Hilliard
 >>=Marc LeBrun
 >>>=Edwin Clark
 >>>>=Cino Hilliard

 >>>>Polynomial numbers of the form 4^n + 4^(n-1) + ... + n mod 2 + 1
 >>> The formula is not clear. I think you mean:
 >>>   (4^n + 4^(n-1) + ... + 4 + 1) + (n mod 2)
 >>> Is this correct?
 > Yes.  This is an interpretation of what I intended.  I was being brief.

Unfortunately, also obscure.  The ellipsis implies there's a pattern 
bridging the 4^(n-1) and the n mod 2 terms, that then carries through to 
the 1 term.  However there is in fact no such pattern, since the n mod 2 is 
simply an additional fudge term, right?  So it doesn't belong in the 
middle.  That's the suggestion.

 > If n is even 1 is the constant term else it is 2.

Yes, of course.

 > To me 4^n = 4^(n-1) ... implies expanding the polynomial to the point of 
4^(n-n) = 1

Do you mean you're expanding 4^n about the point 1?  Whatever, it's 
immaterial.  We all just want the OEIS entries to be as clear as possible, 
right?

 > The addition n mod 2 is MY arbitrary choice to create a sequence of a 
particular character..

Sure, whatever interests you is fine.

 >> I agree that this is a much better way to write it--that mod in the
 >> penultimate term is confusing (as is the adjective "polynomial").
 > The n mod 2  + 1 is MY formula to toggle between 1 and 2 depending on 
whether n is even or odd.
 > I thoughrt it rather clever to avoid a if clause. Eg.,
 > Instead of my  Polynomial numbers of the form  4^n + 4^(n-1) + ... + n 
mod 2 + 1 try
 > Polynomial numbers of the form  4^n + 4^(n-1) + ... + r  where r = 1 if 
n is even or r=2 if n is odd.
 > 6 to .5 DZ to the other.

Using mod in this way is of course quite common.

The suggested improvement is simply to not stick it arbitrarily into the 
middle of the expansion of (4^(n+1)-1)/3.

 > To me polynomial means "many names" or many terms. I see no problem 
using poloynomial in this context.

'When *I* use a word,' Humpty Dumpty said, in a rather scornful tone,' it 
means just what I choose it to mean, neither more nor less.'

'The question is,' said Alice, 'whether you *can* make words mean so many 
different things.'

 >> And the repunit part of course sums to the expression (4^(n+1)-1)/3
 >> (ie (b^(n+1)-1)/(b-1) for other values of 4)
 > Does the repunit create my sequence?

Yes.  Your sequence is just the base 4 repunits plus (n mod 2).  This 
basically just divides everything into a vanilla repunits case and a 
repunits+1 case.

 >>> http://primes.utm.edu/glossary/page.php?sort=GeneralizedRepunit
 > Thanks. I did a google search for 4^n, 4^(n-1) etc without meaningful 
hits. Must have missed it.

You're welcome!  The repunits stuff is somewhat base independent, so the 4 
isn't helpful.  Check out the link Edwin Clark cites, it has a lot of 
related stuff.

 >> explain why the lists for b=4, 9, 16 and 25 are so short.)" which
 >> suggests that there may be a theorem for square b.
 > I will check it out.

It's actually easy top work out, and quickly shows the factorization.  I 
don't know why the repunits web page is so coy.

Specifically: for b a square either every term is composite (b odd) or 
every but 2 (b even, including of course 4).

 > Hey, I am just having fun! Recall Fermat was shooting when he assumed 2^32+1
 > was prime. It was beleived for some 95 years to be true  by some of the 
greatest minds
 > until Euler factored it in his head to be  641*6700417. Imagine that. 
any 8th grader could have done
 > it  by trial division of the 119 primes <= 641. Say 3 division 
(checking) = 5 minutes. 119*5 =  595
 > minutes =~ 10 hours! spread it. a class of 20 could have done int in 1/2 
hour!
 > This seems remarkable to me because for 95 years so many thought it not 
to the point of not
 > questioning or doing it.

Not quite the same thing.  Being a large prime is remarkable; being 
composite isn't (in inverse measure!<;-).

Here, rather than spending 95 years factoring large numbers, the suggestion 
is to spend 5 minutes doing a little algebra to see that this is a nearly 
immediate property of the construction.

 > These are the primes involved in the project.

What exactly *is* the project?  How does it overlap with the repunit stuff?

Thanks!