A064539 All...are divisible by 3 but why?

[zak seidov]

To:
 Simon Nickerson, David Wilson
 and all you seqfan gurus,

 thanks a lot for amazingly 
 prompt (and clear) replies.
 I was so much sure that 
 it's trivial (though not known to me), 
 that i submitted A109216, 
 (see partucularly %C line),
 thanks again, zak

%I A109216
%S A109216 
3,17,3,3,593,3,3,32993,3,3,2097593,3,3,73,3,3,8589935681,3,3,17,3,3,11,3,3,83,3,3,11,3,3,17,3,3,857,3,3,71329,3,3,59,3,3,19,3,3,11,3,3,17
%N A109216 Smallest factor of 2^(2n+1)+(2n+1)^2.
%C A109216 For
n={0,1,4,7,10,16,1003,1063,1879,14677,17326,28642,49534}

we have the only  primes of the form
2^(2n+1)+(2n+1)^2: A064539. Cases 
2n+1 == 1,2 (mod 3) are trivial and may be omitted.
%t A109216
Table[With[{k=2},FactorInteger[n^k+k^n]][[1,1]],{n,1,100,2}]
%Y A109216 A064539, A109215.
%O A109216 0
%K A109216 ,base,nice,nonn,
%A A109216 Zak Seidov (zakseidov at yahoo.com), Jun 24
2005


--- Simon Nickerson <simonn at for.mat.bham.ac.uk> wrote:

> On Thu, 23 Jun 2005, zak seidov wrote:
> > Dear seqfans,
> > A064539= (*Numbers n such that 2^n+n^2 is
> >
>
prime*){1,3,9,15,21,33,2007,2127,3759,29355,34653,57285,99069}All
> > except first are divisible by 3 but why? zak
> 
> Let f(n) = 2^n+n^2. Each integer n can be written
> uniquely as n=6k+l with
> 0<=l<=5.
> 
> f(n) = f(6k+l)
>      = 2^(6k+l) + (6k+l)^2
>      = 4^k 2^l + l^2 (mod 6)
> 
> Now if k>1, then 4^k = 4 (mod 6) (4*4=16=4 (mod 6),
> then use induction),
> so the residue of f(6k+l) mod 6 is 4,3,2,5,2,3 for
> l=0..5. If k=1, then
> the residues turn out to be 1,3,2,5,2,3.
> 
> Now if p>3 is prime, then p is congruent to 1 or 5
> modulo 6. It is clear
> that f(n)>3 if n>1. So if f(n) is prime, n must be
> equal to 1 or congruent
> to 3 modulo 6 (i.e. odd and divisible by 3).
> 
> -- 
> | Simon Nickerson           
> http://web.mat.bham.ac.uk/S.Nickerson
> | simonn at maths.bham.ac.uk                    School
> of Mathematics
> |                                     The University
> of Birmingham
> |_________________________________   Edgbaston,
> Birmingham B15 2TT
> 
> 
> 




		
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