The 4^n Polynomial conjecture

[cino hilliard]

Hi,

I recently submitted the following.

%I A000001
%S A000001
6,21,86,341,1366,5461,21846,87381,349526,1398101,5592406,22369621,89478486,
357913941,1431655766,5726623061,22906492246,91625968981,366503875926,
1466015503701,5864062014806,23456248059221,93824992236886,375299968947541,
1501199875790166,6004799503160661,24019198012642646,96076792050570581,
384307168202282326,1537228672809129301
%N A000001 Polynomial numbers of the form 4^n + 4^(n-1) + ... + n mod 2 + 1

%C A000001 The 4^n Polynomial conjecture:
Base 4 polynomial numbers with constant term = n mod 2 + 1 are composite.
Or, 4^n + 4^(n-1) + ... + n mod 2 +1 is composite for all n > 0. For n=0 
we
have 4^0+1 = 2 prime. Also this appears to be true for powers of 4 in 
general.
While 32 is not a power of 4, 32^n + 32^(n-1) + ... + n mod 2 + 1 is 
composite
also.
%o A000001 (PARI) trajpolypn(n1) =
             {
             for(x1=1,n1,
             y1 = polypn(4,x1);
             print1(y1",")
             )
             }

polypn(n,p) =
          {
           x=n;
            if(p%2,y=2,y=1);
            for(m=1,p,
            y=y+x^m;
                );
           return(y)
           }

Maybe some one can help me prove or disprove the case for 4, 4^k and the 
oddball 32.
Also 8^n +... produces only the prime 73. It look like another situation of 
probability - the
larger the numbers get the scarcer the primes.


Here are a few factorizations for the odd values.

n seq      factors ([p,power;prime,power;..])
2 21 [3, 1; 7, 1]
4 341 [11, 1; 31, 1]
6 5461 [43, 1; 127, 1]
8 87381 [3, 2; 7, 1; 19, 1; 73, 1]
10 1398101 [23, 1; 89, 1; 683, 1]
12 22369621 [2731, 1; 8191, 1]
14 357913941 [3, 1; 7, 1; 11, 1; 31, 1; 151, 1; 331, 1]
16 5726623061 [43691, 1; 131071, 1]
18 91625968981 [174763, 1; 524287, 1]
20 1466015503701 [3, 1; 7, 2; 43, 1; 127, 1; 337, 1; 5419, 1]
22 23456248059221 [47, 1; 178481, 1; 2796203, 1]
24 375299968947541 [11, 1; 31, 1; 251, 1; 601, 1; 1801, 1; 4051, 1]
26 6004799503160661 [3, 3; 7, 1; 19, 1; 73, 1; 87211, 1; 262657, 1]
28 96076792050570581 [59, 1; 233, 1; 1103, 1; 2089, 1; 3033169, 1]
30 1537228672809129301 [715827883, 1; 2147483647, 1]
32 24595658764946068821 [3, 1; 7, 1; 23, 1; 67, 1; 89, 1; 683, 1; 20857, 1; 
5994
79, 1]

Certainly the Mersenne primes appear as factors in some cases but I don't 
see a  pattern
that would deduce a formula.  I guess we could build a sequence of factors 
of terms of the
sequence.

my program just completed testing for n up to 10000 without finding a prime.
(07:01) gp > trajpolypn(10000,4)
(07:35) gp >

Have fun,
Cino

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