[ November 15, 2002 ] [ Last update [ June 24, 2023 ]
A free forum for Gigantic Primes

Last week my PrimeForm program pfgw.exe told me that
10⁹⁹⁹⁹ + 33603
is very probably the smallest gigantic prime.

Norman Luhn (email) let me know [ June 22, 2023 ]
he found the smallest gigantic prime triplet (10000 digits).
Visit his page at https://pzktupel.de/smx0000.php
10⁹⁹⁹⁹ + 194011932117 + d, d = 0, 4, 6

Armed with this knowledge I went to the internet with this number
only to find out that Daniel Heuer already created a list with the
smallest gigantic primes up to 10⁹⁹⁹⁹ + d (from 08/2000 onwards).
http://www.primenumbers.net/prptop/prptop.php
In total he collected 41 gigantic primes with d < 1,000,000.

The sequence thus continues as follows
55377, 70999, 78571,  97779 , 131673, 139579, 236761, 252391,
282097, 333811, 342037, 355651, 359931, 425427, 436363, 444129,
473143, 479859, 484423, 515787, 543447, 680979, 684273, 709053,
709431, 780199, 781891, 788527, 813019, 829833, 835183, 842527,
865051, 881493, 904549, 906009, 909837, 920199, 950791, 982233

The largest titanic prime is 10⁹⁹⁹⁹ – 11333.

Note that all these numbers are Probable Primes (PRP) for now.
Proving them prime is currently still not possible. Or am I wrong ?
I tried in vain using the deterministic parameters of PrimeForm.

C:\PrimeForm>pfgw -q10^9999+33603 -tp
PFGW Version 20010212.Win_Dev (Beta software, 'caveat utilitor')

Primality testing 10^9999+33603 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 5+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 0.10%
10^9999+33603 is Lucas PRP! (238.050000 seconds)

C:\PrimeForm>pfgw -q10^9999+33603 -tm
PFGW Version 20010212.Win_Dev (Beta software, 'caveat utilitor')

Primality testing 10^9999+33603 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 3
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 0.16%
10^9999+33603 is PRP! (208.500000 seconds)

The first titanic primes can be found in Sloane's
OEIS database under A074282.

The fifth term is special as the d value ( 97779) is palindromic !
( ps. squaring 97779 results in this nice pandigital 9560732841 )

Samuel Yates coined the name titanic and gigantic prime.
Exist there already consensus around the name for
numbers of the size 10⁹⁹⁹⁹⁹ ?
Gargantuan Primes ? (source 1)
Megaprimes are primes with at least a million digits.

Can you discover the first gigantic semiprime ?

Please read Jens Kruse Andersen's work on this issue.

Can you discover the first gigantic twin prime (i'll settle for PRP) ?

Please read Anand Nair's message.

Can you discover the first gigantic palprime (PRP is OK) ?

Please read about Daniel Heuer's efforts.

What more is there to say about gigantic primes ?

[ November 22, 2002 ]
Daniel Heuer found the first gigantic palprime
using pfgw and an ABC2 file of the form

10²ⁿ + a*10ⁿ + 101*b*10^n-1 + 10001*c*10^n-2 +
1000001*d*10^n-3 + 100000001*e*10^n-4 + 1
a: from 0 to 9
b: from 0 to 9
c: from 0 to 9
d: from 0 to 9
e: from 0 to 2

The first palprimes for
n = 5 10¹⁰ + 5*10⁵ + 1, 11 digits
n = 50 10¹⁰⁰ + 303*10⁴⁹ + 1, 101 digits
n = 500 10¹⁰⁰⁰ + 81918*10⁴⁹⁸ + 1, 1001 digits
n = 5000 10¹⁰⁰⁰⁰ + 222999222*10⁴⁹⁹⁶ + 1, 10001 digits

This palindrome is 'prime' and not 'probable prime'
because its form is 1(0)₄₉₉₅edcbabcde(0)₄₉₉₅1
and a N-1 method to proof it is available.

A very nice palprime you found there Daniel !

[ November 22, 2002 ]
Jean Claude Rosa very much likes the expression
gargantuan prime but for researching such numbers
he is currently overflow-ing !!

[ November 25, 2002 ]
To complete this WONplate 144, Daniel Heuer
searched also the largest palprime for some
given sizes with an ABC2 file and pfgw.exe.

10⁹ - 272*10³ - 1
10⁹⁹ - 101*10⁴⁸ - 1
10⁹⁹⁹ - 25352*10⁴⁹⁷ - 1
10⁹⁹⁹⁹ - 3927293*10⁴⁹⁹⁶ - 1

All are proved with a N+1 method.

The ABC2 file is of the form :

10^2n-1 – a*10ⁿ – 101*b*10^n-1 – 10001*c*10^n-2 –
1000001*d*10^n-3 – 100000001*e*10^n-4 – 1

[ February 15, 2005 ]
Jens Kruse Andersen found the first prime
of the form 10⁹⁹⁹⁹ + palprime.
The first PRP is 10⁹⁹⁹⁹ + 38719591783, found with PrimeForm.
10⁹⁹⁹⁹ + d was found pretty straightforward in two stages.
First the smallest palprimes were found with ABC2 like Daniel Heuer.
Then pfgw -f on ABC 10^9999+$a, with the list of small palprimes.

[ February 15 & 17, 2005 ]
Jens Kruse Andersen found a small gigantic semiprime
but he doesn't know whether it is the first.

10⁹⁹⁹⁹+1253 is (probably) semiprime
with factors 3 and (10⁹⁹⁹⁹+1253)/3.
I am not aware of a method to test if a composite without a known
factor is a semiprime, so I searched factors and prp'ed the cofactor
with PrimeForm when exactly one factor was found.
10⁹⁹⁹⁹ + k was sieved to 2*10^11.
The first PRP cofactor was (10⁹⁹⁹⁹+1253)/3.
The sieve found no factor for 40 k values below 1253, all composite.
gmp-ecm 5.1 later factored 10 of them with B1=2000 in 30 curves:
(k, factor of 10⁹⁹⁹⁹+k) =
(21,2280935345466254629), (63,1794963036278813),
(279,372106755207751), (303,508074511546073),
(411,6385496027501), (1047,134737627110318853),
(1083,1514639191399), (1087,3655949936867),
(1129,1614742379519), (1227,508074107262121).
All cofactors are composite.
I don't know whether any of the 30 remaining k give semiprimes:
19, 37, 87, 97, 121, 193, 207, 213, 273, 283,
327, 427, 439, 543, 679, 693, 721, 789, 811, 843,
867, 891, 949, 999, 1039, 1053, 1081, 1089, 1231, 1237.

Eliminating all 30 semiprime candidates below 10⁹⁹⁹⁹+1253
seems infeasible. Some of them would probably be factored with
more effort. But statistically some of the 30 numbers will have
their smallest factor too large to find with current methods.
I think the best chance to settle the issue currently would be to
hit a factor with a prime cofactor in one of the smallest numbers,
so all the larger numbers don't have to be factored.

It may be a different matter with CYF NO. 11 at
http://www.shyamsundergupta.com/canyoufind.htm
Any factor attempt on 10⁹⁹⁹+13 might suddenly
succeed and that would solve the problem.
( Update CYF 11 dd. 05-10-2014 :
10⁹⁹⁹+139 is now proven to be the smallest titanic semiprime! )
It is of course also possible that the smallest factor is too
large to be found. Paul Zimmermann gave it a good try.

[ July 13, 2013 ]
Anand Nair found a smallest gigantic twin prime but...

After several weeks of (on and off) computation, I've found
the smallest twin PRP with exactly 10000-digits.

10⁹⁹⁹⁹ + 2421018649 & + 2 are twin PRPs.

but after sending congratulations Anand Nair replied ...

[ July 15, 2013 ]
Well, the congratulations are rather short lived :(
A quick search on the internet with my k value turned up this:
http://tech.groups.yahoo.com/group/primeform/message/9912
It was found 3.5 years ago by Dirk Augustin !
I am only the rediscoverer.