## From bad to worse...from good to Angelic

Expand Messages
• I actually didn t find bad primes defined in any number theory way... they were associated with a Calabi-Yau based L-function in a paper with no real good
Message 1 of 1 , May 23, 2008
I actually didn't find "bad primes" defined in any number theory way...
they were associated with a Calabi-Yau based L-function in
a paper with no real good definition.
So I did a web search and an OEIS search and some experimentation.
A little thought of the geometric average basis of the defintion gave
me an higher order "bad" I call "worse".
An a good I call "Angel".

%I A000001
%S A000001 19, 43, 47, 61, 73, 79, 83, 89, 109, 113, 137, 139, 157,
167, 181, 197, 199,
211, 229, 233, 239, 241, 271, 281, 283, 293, 313, 317, 353, 359, 383, 389,
401, 439, 443, 449, 463, 467, 503, 509, 521, 523
%N A000001 Worse primes<-bad primes (A130903):
If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2] < 0,
Prime[n]]
%F A000001 a(n) = If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2] < 0,
Prime[n]]
%t A000001 Flatten[Table[If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2]
< 0,Prime[n], {}], {n, 3, 100}]]
%Y A000001 Cf. A130903
%O A000001 1
%K A000001 ,nonn,
%A A000001 Roger Bagula and Gary W. Adamsom (rlbagulatftn@...),
May 23 2008
RH
RA 192.20.225.32
RU
RI

%I A000001
%S A000001 5, 7, 11, 13, 17, 23, 29, 31, 37, 41, 53, 59, 67, 71, 97,
101, 103, 107,
127, 131, 149, 151, 163, 173, 179, 191, 193, 223, 227, 251, 257, 263, 269,
277, 307, 311, 331, 337, 347, 349, 367, 373, 379, 397, 409, 419, 421, 431,
433, 457, 461, 479, 487, 491, 499, 541
%N A000001 Angel primes<-good primes (A028388):
If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2] > 0,
Prime[n]]
%C A000001 This sequence set is a next level higher of good and bad
primes.
The bad primes come from modular forms and L-Series theory.
http://en.wikipedia.org/wiki/Arithmetic_of_elliptic_curves:
Quote:
"Reduction mod p
Reduction of an abelian variety A modulo a prime ideal of (the
integers of) K - say, a prime number p - to get an abelian variety Ap
over a finite field, is possible for almost all p. The 'bad' primes,
for which the reduction degenerates by acquiring singular points, are
known to reveal very interesting information. As often happens in
number theory, the 'bad' primes play a rather active role in the theory.
Here a refined theory of (in effect) a right adjoint to reduction mod
p - the NÃ©ron model - cannot always be avoided. In the case of an
elliptic curve there is an algorithm of John Tate describing it.
For abelian varieties such as Ap, there is a definition of local
zeta-function available. To get an L-function for A itself, one takes
a suitable Euler product of such local functions; to understand the
finite number of factors for the 'bad' primes one has to refer to the
Tate module of A, which is (dual to) the Ã©tale cohomology group
H1(A), and the Galois group action on it. In this way one gets a
respectable definition of Hasse-Weil L-function for A. In general its
properties, such as functional equation, are still conjectural - the
Taniyama-Shimura conjecture was just a special case, so that's hardly
surprising.
It is in terms of this L-function that the conjecture of Birch and
Swinnerton-Dyer is posed. It is just one particularly interesting
aspect of the general theory about values of L-functions L(s) at
integer values of s, and there is much empirical evidence supporting it."
%D A000001 http://en.wikipedia.org/wiki/Arithmetic_of_elliptic_curves
%F A000001 a(n) = If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2] <>0,
Prime[n]]
%t A000001 Flatten[Table[If[Prime[n]^2 - Prime[n + 2]*Prime[n - 2]
> 0,Prime[n], {}], {n, 3, 100}]]
%Y A000001 Cf. A130903,A028388
%O A000001 1
%K A000001 ,nonn,
%A A000001 Roger Bagula and Gary W. Adamsom (rlbagulatftn@...),
May 23 2008
RH
RA 192.20.225.32
RU
RI

--
Respectfully, Roger L. Bagula