Paul Underwood
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person 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, p367, p372, c70, c73
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 07/19/2014)
Active primes:on current list: 8 (unweighted total: 14), rank by number 107
Total primes: number ever on any list: 1521.28 (unweighted total: 2978)
Production score: for current list 46 (normalized: 29), total 46.8134, rank by score 117
Largest prime: 3 · 23136255 - 1 ‏(‎944108 digits) via code L256 on 03/08/2007
Most recent: V(94823) ‏(‎19817 digits) via code c73 on 05/26/2014
Entrance Rank: mean 76933.21 (minimum 12, maximum 103515)

Descriptive Data: (report abuse)
Probable Prime Tests:
Trinomial Test
Define F(a,m,r) = am-ar-1
where
a,m,r in N
a > 1
m > 2
m>r>0
except F(2,m,m-1)

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

If aF=a modulo F then amF-arF-1 = 0 modulo F.
The roots satisfy: sum(rootsk) = sum(rootsFk) modulo F for all k.
(symmetric pseudoprime)

For the specific example F(a,3,1) = a3-a-1
the roots satisfy: sum(rootsF) = 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 Test
05 Jun 2005: f=a2-2 tested with 5 Miller-Rabin rounds for a base-a PSP ; none found for all odd a from 3 to 10^11 (3,809,286,968 PRPs.)
26 Apr 2006: f=a2-2 further tested by Carlos Eduardo to a=344,360,000,003 (12,480,999,468 PRPs.)
29 Nov 2013: tested with (L+2)^(f+1)==5 (mod f, L^2+1) for odd a < 10^12 (34,788,375,185 PRPs.)
Unifying Test
For integers a>1, s>=0, all r>0, all t>0, odd and irreducible {a^s\times\prod{(a^r-1)^t}}-1 is a-PRP, 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 and Q=5 Lucas Test
N>5 and any P such that gcd(P,n)==1 and
KroneckerSymbol(P^2-4*5,N)==-1 and KroneckerSymbol(P^2+4*5,N)==-1 and
L^(N+1)==5 (mod N, L^2-P*L+5) and L^(N+1)==-5 (mod N, L^2-P*L-5).
6-Selfridge A-2 and A+2 Fermat-Lucas Test
N>5 coprime to 30, any A such that
JacobiSymbol((A-2)^2-4,N)==-1 and JacobiSymbol((A+2)^2-4,N)==-1, test
(A-2)^N==A-2 (mod N) and (A+2)^N==A+2 (mod N) and
L^(N+1)==1 (mod N, L^2-(A-2)*L+1) and
L^(N+1)==1 (mod N, L^2-(A+2)*L+1).
Verified for N < 1.16*10^8 and for Carmichael numbers < 2^32.
6-Selfridge A-1 and A+1 Fermat-Lucas Test
N>5 coprime tp 210*A, any A such that
JacobiSymbol((A-1)^2-4,N)==-1 and JacobiSymbol((A+1)^2-4,N)==-1, test
(A-1)^N==A-1 (mod N) and (A+1)^N==A+1 (mod N) and
L^(N+1)==1 (mod N, L^2-(A-1)*L+1) and
L^(N+1)==1 (mod N, L^2-(A+1)*L+1).
Verified for N < 1.16*10^8 and for Carmichael numbers < 2^32.
Derived 5-Selfridge A-2 and A+2 Fermat-Lucas Test
N>5 coprime to 30, find A such that
JacobiSymbol((A-2)^2-4,N)==-1 and JacobiSymbol((A+2)^2-4,N)==-1, test
4^N==4 (mod N)
L^(N+1)==1 (mod N, L^2-(A-2)*L+1) and
L^(N+1)==1 (mod N, L^2-(A+2)*L+1).
Liar: N=105809903; A=15164718
Derived 5-Selfridge A-1 and A+1 Fermat-Lucas Test
N>5 coprime to 210*A find x such that JacobiSymbol((A-1)^2-4,N)==-1 and JacobiSymbol((A+1)^2-4,N)==-1, test
2^N==2 (mod N) and
L^(N+1)==1 (mod N, L^2-(A-1)*L+1) and
L^(N+1)==1 (mod N, L^2-(A+1)*L+1).
Liar: N=2499327041; A=20003797
Plus and Minus Conjecture
For prime p>5 with A such that KroneckerSymbol((A-2)^2-4,p)==-1 and KroneckerSymbol((A+2)^2-4,p)==-1 implies 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 N>5 coprime to 30, 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 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 < 6.7*10^14.
L-2 and L+2 Test
N>1, for any integer x such that KoneckerSymbol(x^2-4,N)==-1, test
(L-2)^(N+1)==5-2*x (mod N, L^2-x*L+1) and
(L+2)^(N+1)==5+2*x (mod N, L^2-x*L+1)
Verified for odd N < 2.7*10^7.
5-Selfridge Fermat-Euler-Lucas Test
N>5 coprime to 30, for any integer x:
gcd(x^3-x,N)==1 and JacobiSymbol(x^2-4,N)==-1, test
(x-2)^((N-1)/2)==JacobiSymbol(x-2,N) (mod N) (Euler) and
(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)
Verified for N< 2.6*10^7 and Carmichael numbers < 2^32.
Links to David Broadhurst's liars: 1, 2, 3, 4, 5 and 6
Quartic Test for L+x^2-2
N coprime to 210, 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, test
(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 < 2.6*10^7.

Quartic Test for L+x+1
Odd N>5, for any x such that
N does not divide x and JacobiSymbol(x^2-4,N)==-1 and gcd(x^2+x,N)==1, test
(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 < 1.3*10^7.

Quartic Test for L+x^2-1
Odd N>7, for any x such that
N does not divide x and JacobiSymbol(x^2-4,N)==-1 and gcd(x^2-1,N)==1, test
(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 < 1.9*10^7.

Quartic Test for L+x^2
Odd N>7, for any x such that
N does not divide x and JacobiSymbol(x^2-4,N)==-1 and gcd(x^2-1,N)==1, test
(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 < 2.1*10^7.

x and 2*x or x^2 or x+2 Double Quadratic Test
N coprime to 30, for any x such that
JacobiSymbol(x^2-4,N)==-1 and gcd(x^3-x,N)==1, test
(L+x)^(N+1)==1+2*x^2 (mod N, L^2-x*L+1) and
(L+A)^(N+1)==1+A^2+x*A (mod N, L^2-x*L+1)
where A = 2*x or x^2 or x+2. Verified for N < 2.5*10^7.

Single Parameter Double Lucas with Double Parameter Double Euler Test
N coprime to 6, for any k and a such that
gcd(k*a,N)==1 and
JacobiSymbol((1+a)^2-4),N)==-1 and JacobiSymbol((1-a)^2-4,N)==-1, test
(k^2*(1-a))^((N-1)/2)==JacobiSymbol(1-a,n) (mod N) and
(k^2*(1+a))^((N-1)/2)==JacobiSymbol(1+a,n) (mod N) and
L^((N+1)/2)== JacobiSymbol(1-a,N) (mod N, L^2+(1+a)*L+1) and
L^((N+1)/2)== JacobiSymbol(1+a,N) (mod N, L^2+(1-a)*L+1).
Verified for N < 1.8*10^7 and for Carmichael numbers < 2^32.
L-1 and L+1 Test
(This is a reformulation of the above, with k=1.)
N odd, for any a such that gcd(a,N)==1 and
JacobiSymbol((a+1)^2-4),N)==-1 and JacobiSymbol((a-1)^2-4,N)==-1, test
(L-1)^(N+1)==1-a (Mod N, L^2-(a+1)*L+1) and
(L+1)^(N+1)==1+a (Mod N, L^2-(a-1)*L+1).
Verified for N < 1.16*10^8 and for Carmichael numbers < 2^32.
Quad Test
N coprime to 6, for any x, a and b such that
gcd(a*b*x,N)==1 and gcd(a^2-b^2,N)==1 and JacobiSymbol(x^2-4,N)==-1, test
(L+a)^(N+1)==1+a^2+x*a and (L-a)^(N+1)==1+a^2-x*a and
(L+b)^(N+1)==1+b^2+x*b and (L-b)^(N+1)==1+b^2-x*b
all (mod N, L^2-x*L+1). Verified for N < 1.4*10^5. David's Liars: 1, 2 and 3 [N,P=x,a,b]
Fermat+Lucas+Frobenius Test
N coprime to 30, for any x such that
gcd(x^3-x,N)==1 and JacobiSymbol(x^2-4,N)==-1 test
(2*x)^(N-1)==1 (mod N) and L^(N+1)==1 (mod N, L^2-x*L+1) and
(L+x)^(N+1)==2*x^2+1 (mod N, L^2-x*L+1). Verified for N < 6.8*10^6.

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