On 07/09/2000 at 18:27:23 GMT [EMAIL PROTECTED] wrote:
Dear Mike and others,
This sequence is in actuality two sequences in one. The
first are primes of the form 2^n - 2^[(n+1)/2] +1 and the other are
primes of the form 2^n + 2^[(n+1)/2] +1, as you have delineated.
I direct you to the following URL which is the home of the Sloane's
On-Line Encyclopedia of Integer Sequences.
http://www.research.att.com/~njas/sequences/
You will find the sequences as numbers A007670 A007671.
Their addresses are:
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?An
um=A007670
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?An
um=A007671
Hope you make use of this vital resource.
Sincerely yours,
Robert G. "Bob" Wilson v
Ph.D. ATP / CFGI
While happily acknowledging that the first 25 of these 33 (so far known)
primes figure in that excellent database, I strongly dispute your assertion
that "this sequence is in actuality two sequences in one". Why do you want to
split the single series
s[n] = (1+i)^n - 1
depending on whether n = 1 or 7 mod 8 (i.e. s[n] is in the right half of the
complex plane) or not?
Aurifeuille pointed out in 1873 [cf. Knuth, Vol. 2, p. 376] the identity
2^(4*m+2) + 1 = (2^(2*m+1) + 2^(m+1) + 1) * (2^(2*m+1) - 2^(m+1) + 1)
and to this day numbers of the form
(2^(2*m+1) + 2^(m+1) + 1) and (2^(2*m+1) - 2^(m+1) + 1)
seem to only figure in the literature as _factors_ of the supposedly more
worthy-of-study Fermats (see Cunningham project, etc.). But this is to miss
entirely the unitary nature of the Gaussian series of which they are the
modulus. My argument is as follows.
Why is M[n] = 2^n - 1 an interesting sequence?
Consider M[b,n] = b^n-1, for b a rational integer.
For all n = 1, this has the factor (b-1). So M[b,n] is certainly composite
(for n = 2) unless this factor is trivial (i.e. a unit +1 or -1), which
happens in just 2 cases:-
b = 2, giving the Mersennes M[n]; b = 0, which is uninteresting.
Now consider G[c,n] = c^n - 1, for c a complex (Gaussian) integer.
For all n = 1, this has the factor (c-1). So G[c,n] is certainly composite
(for n = 2) unless this factor is trivial (i.e. a unit +1, -1, +i or -i),
which happens now in 4 cases:-
c = 2, giving the Mersennes, M[n];
c = 0, which is uninteresing;
c = 1+i, giving the new series we are talking about, s[n] = (1+i)^n - 1;
c = 1-i, giving the complex conjugate of this series.
One might now naturally be drawn to investigate H[c,n] = c^n - i, for complex
c, but, remarkably, we get no new candidate primes, just some old friends
again:
For all n = 1, H[c,n] has the factor (c-i), and so is composite (for n = 2)
unless this factor is trivial (i.e. a unit +1, -1, +i or -i), which happens
in 4 cases:-
c = 1+i, giving the series (1+i)^n - i, which encompasses all the Fermats
except F0, plus those M[n] and s[n] with n = +1 mod 4;
c = -1+i, giving the series (-1+i)^n - i, which yields all the Fermats except
F0, plus those M[n] with n = -1 mod 4 and those s[n] with n = +1 mod 4;
c = i+i, giving the series (2*i)^n - i, which covers all the Fermats except
F0 F1, plus those M[n] with n = +1 mod 4;
c = 0, which is uninteresing.
Are you not convinced that we have here just one (rather beautiful) new
series?
An appeal to all you guys out there with lots of CPU cycles: let's find the
34th, 35th... prime terms - there's no reason other than machine time why the
40th (e.g.) should not become the largest known prime. Send an email to
[EMAIL PROTECTED] to receive a range of exponents to check out.
Mike Oakes
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