"212288 - 212224 - 1 + 264 · (floor(212158 · Pi) + 7425765)"
At this site we maintain a list of the 5000 Largest Known Primes which is updated hourly. This list is the most important PrimePages database: a collection of research, records and results all about prime numbers. This page summarizes our information about one of these primes.
This prime's information:
| Description: | "212288 - 212224 - 1 + 264 · (floor(212158 · Pi) + 7425765)" |
|---|---|
| Verification status (*): | PRP |
| Official Comment (*): | ECPP, Sophie Germain (2p+1) |
| Unofficial Comments: | This prime has 1 user comment below. |
| Proof-code(s): (*): | c9 : Kivinen, Primo |
| Decimal Digits: | 3700 (log10 is 3699.05658672) |
| Rank (*): | 97620 (digit rank is 2) |
| Entrance Rank (*): | 29467 |
| Currently on list? (*): | no |
| Submitted: | 4/1/2003 |
| Last modified: | 2/4/2026 10:59:16 UTC |
| Database id: | 64849 |
| Blob database id: | 93 |
| Status Flags: | Verify |
| Score (*): | 29.3523 (normalized score 0) |
Description: (from blob table id=93)
From RFC 2412Classical Diffie - Hellman Modular Exponentiation Groups
The primes for groups 1 and 2 were selected to have certain properties. The high order 64 bits are forced to 1. This helps the classical remainder algorithm, because the trial quotient digit can always be taken as the high order word of the dividend, possibly + 1. The low order 64 bits are forced to 1. This helps the Montgomery - style remainder algorithms, because the multiplier digit can always be taken to be the low order word of the dividend. The middle bits are taken from the binary expansion of pi. This guarantees that they are effectively random, while avoiding any suspicion that the primes have secretly been selected to be weak.
Because both primes are based on pi, there is a large section of overlap in the hexadecimal representations of the two primes. The primes are chosen to be Sophie Germain primes (i.e., (P - 1)/2 is also prime), to have the maximum strength against the square - root attack on the discrete logarithm problem.
The starting trial numbers were repeatedly incremented by 2^64 until suitable primes were located.
Because these two primes are congruent to 7 (mod 8), 2 is a quadratic residue of each prime. All powers of 2 will also be quadratic residues. This prevents an opponent from learning the low order bit of the Diffie - Hellman exponent (AKA the subgroup confinement problem). Using 2 as a generator is efficient for some modular exponentiation algorithms. [Note that 2 is technically not a generator in the number theory sense, because it omits half of the possible residues mod P. From a cryptographic viewpoint, this is a virtue.]
Archival tags:
There are certain forms classed as archivable: these prime may (at times) remain on this list even if they do not make the Top 5000 proper. Such primes are tracked with archival tags.
- Elliptic Curve Primality Proof (archivable *)
- Prime on list: no, rank 714
Subcategory: "ECPP"
(archival tag id 175134, tag last modified 2026-01-31 06:37:12)
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Verification data:
The Top 5000 Primes is a list for proven primes only. In order to maintain the integrity of this list, we seek to verify the primality of all submissions. We are currently unable to check all proofs (ECPP, KP, ...), but we will at least trial divide and PRP check every entry before it is included in the list.
field value prime_id 64849 person_id 9 machine Using: Digital Ocean Droplet what prp notes PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 1139165225...5900159999 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_1100000004142070779.txt
trial
Running N-1 test using base 13
Generic modular reduction using generic reduction FMA3 FFT length 1280 on A 12288-bit number
Running N+1 test using discriminant 19, base 1+sqrt(19)
Generic modular reduction using generic reduction FMA3 FFT length 1280 on A 12288-bit number
Calling N+1 BLS with factored part 1.11% and helper 0.01% (3.35% proof)
1139165225...5900159999 is Fermat and Lucas PRP! (0.6440s+0.0006s)
[Elapsed time: 5.00 seconds]
Helper File:
2
3
5
596164171
3548238587modified 2026-02-04 10:59:16 created 2026-02-04 10:59:11 id 187723
field value prime_id 64849 person_id 9 machine Using: Digital Ocean Droplet what prp notes PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 1139165225...5900159999 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_1100000004142070779.txt
trial
Running N-1 test using base 13
Generic modular reduction using generic reduction FMA3 FFT length 1280 on A 12288-bit number
Running N+1 test using discriminant 19, base 1+sqrt(19)
Generic modular reduction using generic reduction FMA3 FFT length 1280 on A 12288-bit number
Calling N+1 BLS with factored part 1.11% and helper 0.01% (3.35% proof)
1139165225...5900159999 is Fermat and Lucas PRP! (0.6548s+0.0011s)
[Elapsed time: 5.00 seconds]
Helper File:
2
3
5
596164171
3548238587modified 2026-02-04 10:59:11 created 2026-02-04 10:59:06 id 187722
field value prime_id 64849 person_id 9 machine Linux P4 2.8GHz what prp notes PFGW Version 20020311.x86_Dev (Alpha software, 'caveat utilitor') Running N-1 test using base 13 Primality testing 1139165225...5900159999 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N+1 test using discriminant 19, base 1+sqrt(19) Running N+1 test using discriminant 19, base 2+sqrt(19) Running N+1 test using discriminant 19, base 3+sqrt(19) Calling N+1 BLS with factored part 0.62% and helper 0.01% (1.87% proof) 1139165225...5900159999 is Fermat and Lucas PRP! (28.750000 seconds) modified 2020-07-07 22:30:47 created 2003-05-27 14:56:12 id 69717