The Prime Pages: Paul Underwood

Paul Underwood
(Another of the Prime Pages' resources)

A titan, as defined by Samuel Yates, is anyone who has found a titanic prime. This page provides data on those that have found these primes. The data below only reflect on the primes currently on the list. (Many of the terms that are used here are explained on another page.)

Proof-code(s):	p6, x43, p98, p102, p62, L97, L158, L256
E-mail address:	paulunderwood@mindless.com
Username:	Underwood	(entry created on 01/18/2000)
Database id:	181	(entry last modified on 09/10/2008)
Active primes:	on current list: 3.7 (unweighted total: 5), rank by number 174
Total primes:	number ever on any list: 93.95 (unweighted total: 123)
Production score:	for current list 46 (normalized: 1239), total 46.7915, rank by score 23
Largest prime:	3 · 2^3136255-1 ‏(‎944108 digits) via code L256 on 03/08/2007
Most recent:	3 · 2^3136255-1 ‏(‎944108 digits) via code L256 on 03/08/2007
Entrance Rank:	mean 11616.20 (minimum 12, maximum 57855)

To link to this page use the following URL: http://primes.utm.edu/bios/page.php?id=181

Descriptive Data: (report abuse)

Conjectures:

Trinomial Conjecture
Define F(a,m,r) = a^m-a^r-1
where
a,m,r in N
a > 1
m > 2
m>r>0
except F(2,m,m-1)

Conjecture: If a^F=a modulo F then F is prime

If a^F=a modulo F then a^mF-a^rF-1 = 0 modulo F.
The roots satisfy: sum(roots^k) = sum(roots^Fk) modulo F for all k.
(symmetric pseudoprime)

For the specific example F(a,3,1) = a³-a-1
the roots satisfy: sum(roots^F) = 0 modulo F
(Perrin pseudoprime)

What is the probability of a Perrin pseudoprime?
What is the probability that composite n satisfies aⁿ=a modulo n? (Click here for answer)
How many tests are expected to refute the conjecture?

If you find a counterexample, please let me know.

8 Aug 2001: f=a³-a-1 tested for all a less than 1.4*10^9 with 15 Miller Rabin rounds
17 Nov 2001: f=a³-a-1 proved for all a less than 10^8
14 Mar 2002: f=a³+a-1 proved for all a less than 10^8
20 Sep 2002: f=a³-a-1 tested for all a less than 10^10+2 with 5 Miller Rabin rounds (527,345,506 PrPs)
01 Jan 2003: f=a³-a-1 tested for all a less than 10^11 with 5 Miller Rabin rounds (4,772,369,646 PrPs)
01 Jan 2003: f=a³-a-1 tested for all a from 10^11 to 223,490,000,000 with 5 Miller Rabin rounds by Michael Angel

Quadratic Conjecture
05 Jun 2005: f=a²-2 tested with 5 Miller-Rabin rounds for base-a PSP ; none found for all odd a from 3 to 10^11 (3,809,286,968 PRPs)
25 Apr 2006 f=a²-2 further tested by Carlos Eduardo to a=344,360,000,003 (12,480,999,468 PRPs.)

Unifying Conjecture
For integers a>1, s>=0, all r>0, all t>0, if odd and irreducible {a^s\times\prod{(a^r-1)^t}}-1 is a-PRP then it is prime, except for the cases a^2-a-1 and a-2 and a-1 and -1.

FLT-type Conjecture
There are no non-zero integer solutions to A*x^n+B*y^n=C*z^n where |A|+|B|+|C|<=n and x,y,z are distinct.

5-Selfridge Q=+-5 Lucasian Conjecture
gcd(P,n)==1 and both kronecker(P^2-+4*5,n)==-1 and both matrices [P,-+5;1,0]^(n+1)==[+-5,0;0,+-5] (mod n) all together imply "n" is prime. (Both +-5 needed.)

6-Selfridge A+-2 Lucasian Conjecture
gcd(30,n)==1 and both kronecker((A+-2)^2-4)==-1 and both (A+-2)^n==(A+-2) (mod n) and both matrices [A+-2,-1;1,0]^(n+1)==[1,0;0,1] (mod n) all together imply "n" is prime. (Both A+-2 needed.)

6-Selfridge A+-1 Lucasian Conjecture
gcd(210*A,n)==1 and both kronecker((A+-1)^2-4)==-1 and both (A+-1)^n==(A+-1) (mod n) and both matrices [A+-1,-1;1,0]^(n+1)==[1,0;0,1] (mod n) all together imply "n" is prime. (Both A+-1 needed.)

I am Paul Underwood and I would like to

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Surname: Underwood (used for alphabetizing and in codes)
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