Paul Underwood

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 reflects 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 ... ... L5037, E5, E6, E8, E12
E-mail address: paulunderwood@mindless.com
Web page:http://arxiv.org/abs/1706.01265
Username Underwood (entry created on 1/18/2000 18:50:53 UTC)
Database id:181 (entry last modified on 10/21/2023 11:27:57 UTC)
Active primes:on current list: 22.4999 (unweighted total: 42), rank by number 34
Total primes: number ever on any list: 2596.62 (unweighted total: 6167)
Production score: for current list 49 (normalized: 65), total 49.1487, rank by score 119
Largest prime: 69 · 26838971 - 1 ‏(‎2058738 digits) via code L5037 on 3/1/2020 23:53:30 UTC
Most recent: (2138937 + 1)/3 ‏(‎41824 digits) via code E12 on 10/24/2023 01:45:50 UTC
Entrance Rank: mean 64632.45 (minimum 12, maximum 103514)

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 Mar 2015: tested with (L+2)^(f+1)==5 (mod f, L^2+1) for odd a < 10^13 (320,120,182,301 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 < 2*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 < 2*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).
Counterexample: 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).
Counterexample: N=2499327041; A=20003797
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 < 2^50.
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 < 10^8.
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.

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 < 10^8.

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 < 10^8.

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^3-x,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 < 2.6*10^7. Counterexample: N=62164241, x=2290208.
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 < 10^8.
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 < 10^8.

L-1 and L+1 Test
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 < 2*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.
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 < 10^8.
Double Fermat+Frobenius Test
N coprime to 30, for any x and y such that
gcd(x^3-x,N)==1 and JacobiSymbol(x^2-4,N)==-1 and
gcd(y^3-y,N)==1 and JacobiSymbol(y^2-4,N)==-1 and gcd(x^2-y^2,n)==1 test
(2*x)^(N-1)==1 (mod N) and (L+x)^(N+1)==2*x^2+1 (mod N, L^2-x*L+1) and
(2*y)^(N-1)==1 (mod N) and (L+y)^(N+1)==2*y^2+1 (mod N, L^2-y*L+1) and
Verified for N < 10^8.
a^2+k Test
For n=a^2+k coprime to 6 and n>42 where k>0 and a>1 and gcd(k^3-k,n*a)==1 find
x such that gcd(x,n)==1 and jacobiSymbol(x^2-4,n)==-1 and test:
(L+a)^(n+1)==a^2+1+a*x (mod n, L^2-x*L+1) and
(L-a)^(n+1)==a^2+1-a*x (mod n, L^2-x*L+1). Verified for n< 1.3*10^6.
Counterexample: [n, a, k, x]=[1432999, 641, 1022118, 381129]
Q only Test
For n coprime to 6 and n>5 find Q : gcd(Q+1,n)==1 and jacobiSymbol(Q,n)==-1 and jacobiSymbol(Q-4,n)==1 and jacobiSymbol(Q+4,n)==1 and test:
Q^((n-1)/2)==-1 (mod n) and x^((n+1)/2)==-1 (mod n, x^2-(Q-2)*x+1) and
x^((n+1)/2)==-jacobiSymbol(-1,n) (mod n, x^2+(Q+2)*x+1). Verified for n < 2.3*10^7
David's counterexamples: {[[1581223211, 1093877615], [13078382623, 6926732822], [21577464971, 6280410804], [33816027671, 12420727435], [56004757043, 39375517400], [65963127163, 38771733269], [89249862731, 63319378549], [92225622499, 91249250716], [97457318983, 76930744545]]}
2*x+1 and 2*x-1 Test
For odd n, find a such that jacobiSymbol(a^2-4,n)==-1 and test
(2*x+1)^(n+1)==5+2*a (mod n, x^2-a*x+1) and
(2*x-1)^(n+1)==5-2*a (mod n, x^2-a*x+1).
Double Parameter Test
For n coprime to 30 find a and b:
jacobiSymbol(a^2-4,n)==-1 and gcd(b^2-4,n)==1 and gcd(a^2-b^2,n)==1 and
gcd((a^3-a)*(b^3-b),n)==1 and gcd(a^3-a,b^3-b)==6 and test:
(b*x+1)^(n+1)==1+b^2+a*b (mod n, x^2-a*x+1) and
(b*x-1)^(n+1)==1+b^2-a*b (mod n, x^2-a*x+1)
David's counterexamples:
{[[107752185411149, 83669262602538, 19154350998482],
[159132028928489, 38034692864164, 56837927663062],
[2544844174968929, 2391864938130562, 575637140842943],
[1484926074920009, 1014048410273000, 421042641324018],
[2333564831617769, 191389787226297, 210732564557270],
[11149098312734969, 3776761935674103, 1265100831985174],
[8353109377649009, 2445829862238038, 3451336874803033],
[4627936813857689, 2353076244593330, 2940176822193232],
[5099517882159749, 3276100765569083, 2684765653980526],
[7308466463154149, 6633660182802507, 5895989769974006]]}
x-Q and x+Q Test
n coprime to 6, find Q: gcd(Q^2-1,n)==1 and jacobiSymbol(1-4*Q,n)==-1 and test
(x-Q)^(n+1)==Q^2 and (x+Q)^(n+1)==(Q+1)^2-1 both (mod n, x^2-x+Q).
Verified for n < 1.8*10^7.
5 Selfridge Q Test
n coprime to 6, find Q: gcd(Q^2-1,n)==1 and jacobiSymbol(1-4*Q,n)==-1 and
jacobiSymbol(Q,n)==-1 and test Q^((n-1)/2)==-1 (mod n)
and x^((n+1)/2)==-1 (mod n, x^2-(1-2*Q)/Q*x+1) and
x^((n+1)/2)==jacobiSymbol(2+Q,n) (mod n, x^2-(5-2*Q)/(2+Q)*x+1).
Verified for n < 5*10^6. Counterexample: n=5073799 and Q=2484110.
David's counterexamples: {[[76537243, 62685438], [156524443, 1148764], [486819343, 352255858], [724361923, 38713830], [1001493799, 859390320], [7445895763, 916298043], [15728383999, 4890179599]]}
Complex test
For n, test (2+x*I)^n == 2+x*I^((n+1)/2) (mod n, x^2+I)
Claim: pseudoprimes exist only for n == 1 mod 8.
Further complex test
To test n, with integer k such that gcd(k,n)==1,
test (2+x)^n == 2+x^(n%(2*k)) (mod n, x^k+1).
Claim: pseudoprimes exist only for n == 1 mod 2*k.
Base x+1 test
To test n, with integer k (>3) such that gcd(k,n)==1 and
n%(2*k)!=1 and n%(2*k)!=2*k-1 test the following:
(x+1)^n == x^(n%(2*k))+1 (mod n, x^k+1)
2.X-Selfridge Composite Test Algorithm Revisited
For N find integer 0<=x<=ceil(N^0.25) 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 < 10^12.
8 Selfridge test
For odd N>6 find a such that gcd(a^3+a,N)==1 and JacobiSymbol((a*I)^2-4,N)==-1
and perform the probable prime test:
(x+1)^(N+1)==2+a*I mod(N, x^2-a*I*x+1) if N==1 mod 4;
(x+1)^(N+1)==2-a*I*x mod(N, x^2-a*I*x+1) if N==3 mod 4
Verified for N < 10^6. (Note: if a is small this test is 8 selfridges.)
Counterexample: [n,a]=[1037623, 134383]
2 Selfridge Baillie-PSW test
For non-square odd N find minimal a>2 such that JacobiSymbol(a^2-4,N)==-1
and perform the probable prime test:
(2*x)^((N+1)/2)==JacobiSymbol(2*(a+2),N)*2 mod(N, x^2-a*x+1)
Verified for N < 2^64
6^r test
For non-square N with gcd(210,N)==1 and let b=6 and a=6^r (any r),
where gcd(a^3-a,N)==1 and JacobiSymbol(a^2-4,N)==-1. Then test:
(b*x)^((N+1)/2)==b*JacobiSymbol(b*(a+2),N) (mod N, x^2-a*x+1)
Verified for N < 1.5*10^14
A Quadratic Frobenius Composite Test with One Free Parameter
Please see this paper.
2^r tests
For non-square n find a=2^r where gcd(6,n)==1, kronecker(a^2+-4,n)==-1,
gcd(a^2+-1,n)==1 and gcd(a^2+-2,n)==1 and then test
(a^2+-4)^((n-1)/2) == -1 (Mod n) and :-
if "+" case: (2*x)^(n+1) == -4 (Mod n, x^2-a*x-1)
if "-" case: (2*x)^((n+1)/2) == 2*kronecker(2*(a+2),n) (Mod n, x^2-a*x+1)
Both verified for n < 5*10^12.
Two Parameter test
For any non-square n find a=2^r and b:
b%n!=0 and gcd(a^2-b,n)==1 and gcd(a^2-2*b,n)==1
and kronecker(a^2-4*b,n)==-1 and test:
(a^2-4*b)^((n-1)/2)==-1 (mod n) and
(2*x)^(n+1)==4*b (mod n, x^2-a*x+b)
Verified for n <= 4502485
Euler-Frobenius Test
For non-square n find b such that Jacobi(1-b,n)==-1
and gcd(4-b,n)==1 and gcd(4-2*b,n)==-1 and then test:
(1-b)^((n-1)/2)==-1 (mod n)
x^(n+1)==b (mod n, x^2-2*x+b)
Verified for n < 10^6
[n,b] = [79786523, 2982537] is a counterexample
Euler-Frobenius 2^r Version
For non-square odd n find a=2^r such that
gcd(a-2,n)==1 and gcd(a-4,n)==1 and kronecker(1-a,n)==-1
and test (1-a)^((n-1)/2)==-1 (mod n) and
x^(n+1)==a (mod n, x^2-2*x+a)
Verified for n < 9*10^7
Euler-Frobenius 2^r Variant
For non-square odd n find a=2^r such that
gcd(a-2,n)==1 and gcd(a-4,n)==1 and kronecker(1-a,n)==-1
and test (1-a)^((n-1)/2)==-1 (mod n) and
2^(n-1)==1 (mod n) and
x^(n+1)==1 (mod n, x^2-(4/a-2)*x+1)
Verified for n < 10^13
x^2-y Test
For odd non-square n find any y=2^r such that
gcd(r-1,n-1)==1 and kronecker(y,n)==-1 and test
2^(n-1)==1 (mod n) and
y^((n-1)/2)==-1 (mod n) and
(x+1)^(n+1)==-y+1 (mod n, x^2-y)
Verified for n < 10^12
Simple Euler-Frobenius Test
For odd n find a < sqrt(n) such that jacobi(a,n)==-1
and test a^((n-1)/2)==-1 (mod n) and
(x+1)^n==-x+1 (mod n, x^2-a)
Reduced Domain Test
Paper (£100 Prize)
Lucas(n,12^r,-12) Test
For any n and r with gcd(6,n) and kronecker(144^r+48,n)==-1
test x^(n+1)==-12 (mod n, x^2-12^r*x-12).
Lucas(n,3^r,-3) Test
For any n with gcd(6,n)==1 and kronecker(9^r+12,n)==-1 and gcd(r-1,n-1)
test x^(n+1)==-3 (mod n, x^2-3^r*x-3). Paper.
Lucas(n,2^r,-2) Test
For any n with gcd(6,n)==1 and kronecker(9^r+8,n)==-1 and gcd(2*r-1,n-1)
test x^(n+1)==-2 (mod n, x^2-2^r*x-2).
TPPPE test
{ kill(x);TPPPE(n)=my(k=2,X=x); while(X==x,k++;X=Mod(Mod(x,n),x^k-x-1)^n); gcd(polcoef(lift(lift(X)),k-1),n)==1&&X^k-X-1==0; }
Cubic Test
{kill(x);TPPPCb(n)=my(b=1,X=Mod(Mod(x,n),x^3-b*x-1)^n);
while(gcd(b,n)!=1||kronecker(4*b^3-27,n)!=1||X==x,b++;X=Mod(Mod(x,n),x^3-b*x-1)^n);
gcd(polcoef(lift(lift(X)),2),n)==1&&X^3-b*X-1==0&&Mod(b,n)^n==b;}
General Test
Paper here

Surname: Underwood (used for alphabetizing and in codes).
Unverified primes are omitted from counts and lists until verification completed.
I am Paul Underwood and I would like to
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