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, p308
E-mail address:	paulunderwood@mindless.com
Web page:	http://www.mersenneforum.org/showpost.php?p=298027&postcount=44
Username:	Underwood	(entry created on 01/18/2000)
Database id:	181	(entry last modified on 04/11/2013)
Active primes:	on current list: 7 (unweighted total: 12), rank by number 118
Total primes:	number ever on any list: 1518.95 (unweighted total: 2973)
Production score:	for current list 46 (normalized: 53), total 46.7915, rank by score 89
Largest prime:	3 · 2^3136255 - 1 ‏(‎944108 digits) via code L256 on 03/08/2007
Most recent:	78613471382 · 2371# + 1 ‏(‎1014 digits) via code p308 on 04/14/2013
Entrance Rank:	mean 80064.08 (minimum 12, maximum 103515)

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
N>5 and gcd(P,N)==1 and both KroneckerSymbol(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
N>5 and gcd(30,N)==1 and both JacobiSymbol((A+-2)^2-4,N)==-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.) Verified for N< 1.09*10^8 and for Carmichael numbers < 2^32.

6-Selfridge A+-1 Lucasian Conjecture
N>5 and gcd(210*A,N)==1 and both JacobiSymbol((A+-1)^2-4,N)==-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.) Verified for N< 1.09*10^8 and for Carmichael numbers < 2^32.

Derived 5-Selfridge A+-2 Lucasian Conjecture
N>5 and gcd(30,N)==1 and both JacobiSymbol((A+-2)^2-4,N)==-1 and 4^N==4 (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.) Refuted: N=105809903; A=15164718

Derived 5-Selfridge A+-1 Lucasian Conjecture
N>5 and gcd(210*A,N)==1 and both JacobiSymbol((A+-1)^2-4,N)==-1 and 2^N==2 (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.) Refuted: N=2499327041; A=20003797

Plus and Minus Conjecture
For prime p>5 with both KroneckerSymbol((A+-2)^2-4,p)==-1 then either both [A+-2,-1;1,0]^((p+1)/2)==[+-1,0;0,+-1] (mod p) or both [A+-2,-1;1,0]^((p+1)/2)==[-+1,0;0,-+1] (mod p) (Both A+-2 needed.)

1st 2.X-Selfridge Composite Test Algorithm
For non-square N>5, such that gcd(30,N)==1, find the minimal integer x>0 where JacobiSymbol(x^2-4,N)==-1 and perform the probable prime test:
(x*L-3)^(N+1)==9-2*x^2 (mod N, L^2-x*L+1). Verified for N< 2.481*10^12.

2nd 2.X-Selfridge Composite Test Algorithm
For non-square N find minimal integer x>=0 where KroneckerSymbol(x^2-4,N)==-1 and perform the probable prime test: (L+2)^(N+1)==2*x+5 (mod N, L^2-x*L+1). Verified for odd N< 10^13.

4.X-Selfridge Conjecture
Non-square N>1 is prime if and only if for any integer x such that
KoneckerSymbol(x^2-4,N)==-1
then both (L+-2)^(N+1)==5+-2*x (mod N, L^2-x*L+1) (Both L+-2 needed.)
Verified for odd N< 10^7.

5-Selfridge Lucasian Conjecture
Non-square N>5, with gcd(30,N)==1, is prime if and only if for any integer x:
gcd(x^3-x,N)==1 and JacobiSymbol(x^2-4,N)==-1
then both (x+-2)^((N-1)/2)==JacobiSymbol(x+-2,N) (mod N) (Euler), and
x^(N-1)==1 (mod N) (Fermat) and L^(N+1)==1 (mod N, L^2-x*L+1) (Lucas)
(Both x+-2 needed.) Verified for N< 2.6*10^7 and Carmichael numbers < 2^32.
Links to David Broadhurst's counterexamples: 1, 2, 3, 4, 5 and 6

Quartic Conjecture for L+x^2-2
Non-square N, coprime to 210, is prime if and only if for any x indivisible by n
and JacobiSymbol(x^2-4,N)==-1 and gcd((x^3-x)*(x^2-2)*(x^2-3),N)==1, then
(L+x^2-2)^N==-L^3+(x^2-2)*L+x^2-2 (mod N, (L^2-x*L+1)*(L^2+x*L+1)).
Verified for N< 9.5*10^6.

Quartic Conjecture for L+x+1
Non-square and odd N>5 is prime if and only if for any x such that
N does not divide x and JacobiSymbol(x^2-4,N)==-1 and gcd(x+1,N)==1, then
(L+x+1)^N==-L^3+(x^2-2)*L+x+1 (mod N, (L^2-x*L+1)*(L^2+x*L+1)).
Verified for N< 6.3*10^6.

Quartic Conjecture for L+x^2-1
Non-square and odd N>7 is prime if and only if for any x such that
N does not divide x and JacobiSymbol(x^2-4,N)==-1 and gcd(x^2-1,N)==1, then
(L+x^2-1)^N==-L^3+(x^2-2)*L+x^2-1 (mod N, (L^2-x*L+1)*(L^2+x*L+1)).
Verified for N< 7.3*10^6.

Quartic Conjecture for L+x^2
Non-square and odd N>7 is prime if and only if for any x such that
N does not divide x and JacobiSymbol(x^2-4,N)==-1 and gcd(x^2-1,N)==1, then
(L+x^2)^N==-L^3+(x^2-2)*L+x^2 (mod N, (L^2-x*L+1)*(L^2+x*L+1)).
Verified for N< 8.3*10^6.

x and 2*x Double Quadratic Conjecture
Non-square N coprime to 30 is prime if and only if for any x such that
JacobiSymbol(x^2-4,N)==-1 and gcd(x^3-x,N)==1, then
(L+x)^(N+1)==1+2*x^2 (mod N, L^2-x*L+1) and
(L+2*x)^(N+1)==1+6*x^2 (mod N, L^2-x*L+1). Verified for N< 4.3*10^6.

x and x^2 Double Quadratic Conjecture
Non-square N coprime to 30 is prime if and only if for any x such that
JacobiSymbol(x^2-4,N)==-1 and gcd(x^3-x,N)==1, then
(L+x)^(N+1)==1+2*x^2 (mod N, L^2-x*L+1) and
(L+x^2)^(N+1)==1+x^3+x^4 (mod N, L^2-x*L+1).Verified for N< 2.2*10^6.

I am Paul Underwood and I would like to

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