## The maximum modulus root of this polynomial is always a prime

Expand Messages
• The Perron number type of elliptical equations suggested this polynimial whose maximum modulus root is always a prime: q^2=p^3 + 2*p^2 - (3 - b[n])*p - b[n]
Message 1 of 1 , May 1, 2005
• 0 Attachment
The Perron number type of elliptical equations suggested this polynimial
whose maximum modulus root is always a prime:
q^2=p^3 + 2*p^2 - (3 - b[n])*p - b[n]
b[n]=3*Prime[n]-Prime[n]^2
It's kind of a cheat as the input sequence depends on the primes as well.
The Second biggest root is mostly Abs[Prime[n]-3] except for the second
term.
w^2=(x-1)*(x-Abs[Prime[n]-3])*(x+Prime[n])
The Roots->{1,0,-3}
seem important.

%I A000001
%S A000001 2, 1, 1, 3, 0, 1, 5, 1, 2, 7, 1, 4, 11, 1, 8, 13, 1, 10, 17, 1, 14, 19, 1,
16, 23, 1, 20, 29, 1, 26, 31, 1, 28, 37, 1, 34, 41, 1, 38, 43, 1, 40, 47, 1,
44, 53, 1, 50, 59, 1, 56, 61, 1, 58, 67, 1, 64, 71, 1, 68, 73, 1, 70, 79, 1,
76, 83, 1, 80, 89, 1, 86, 97, 1, 94, 101, 1, 98, 103, 1, 100, 107, 1, 104,
109, 1, 106, 113, 1, 110, 127, 1, 124, 131, 1, 128, 137, 1, 134, 139, 1, 136,
149, 1, 146, 151, 1, 148, 157, 1, 154, 163, 1, 160, 167, 1, 164, 173, 1, 170,
179, 1, 176, 181, 1, 178, 191, 1, 188, 193, 1, 190, 197, 1, 194, 199, 1, 196,
211, 1, 208, 223, 1, 220, 227, 1, 224, 229, 1, 226, 233, 1, 230, 239, 1, 236,
241, 1, 238, 251, 1, 248, 257, 1, 254, 263, 1, 260, 269, 1, 266, 271, 1, 268,
277, 1, 274, 281, 1, 278, 283, 1, 280, 293, 1, 290, 307, 1, 304, 311, 1, 308,
313, 1, 310, 317, 1, 314, 331, 1, 328
%N A000001 Absolute values of the three roots of p^3 + 2*p^2 - (3 - b[n])*p - b[n]:
b[n]=3*Prime[n]-Prime[n]^2
%C A000001 The maximum modulus root of this polynomial is always a prime.
%F A000001 b[n]=3*Prime[n]-Prime[n]^2
a(n) = Abs[root[i,m]]/.p^3 + 2*p^2 - (3 - b[n])*p - b[n]
%t A000001 b= Table[3* Prime[n] - Prime[n]^2, {n, 1, Digits}]
a = Abs[Flatten[Table[p /. Solve[p^3 + 2*p^2 - (3 - b[[n]])*p - b[[n]] == 0, p][[i]], {n, 1, Digits}, {i, 1, 3}]]]
%O A000001 1
%K A000001 ,nonn,
%A A000001 Roger L. Bagula (rlbagulatftn@...), May 01 2005
RH
RA 69.233.150.223
RU
RI

--
Roger L. Bagula email: rlbagula@... or rlbagulatftn@...