# 2 · 7709976 + 2 · 7211441 + 1

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: 2 · 7709976 + 2 · 7211441 + 1 PRP [none] This prime has 1 user comment below. CH9 : Zhou, OpenPFGW, CHG 600000   (log10 is 599999.62708716) 4911 (digit rank is 1) 3801 yes 2/18/2023 19:28:03 UTC 5/20/2023 20:59:19 UTC 135387 Verify 45.065 (normalized score 1.0213)

Lei Zhou writes (18 Feb 2023):  (report abuse)
 This is a balanced base 7 prime with 3 non-zero digits. p-1=2^4*7^211441*11*191*Phi[199414,7]*Phi[997070,7] Using prime number 2,7,11,191 as helper, OpenPFGW provides that it is a Fermat and Lucas PRP: Using F=2^4*7^211441*11*191 and G=6, CHG pari script proved that this is a prime number. (full CHG output is too long to be posted here. Verification is the following) The result is certified by David Broadhurst's verifier chgcertd.gp. Testing a PRP called "cp_709976_+7_211441_c2+1.in". Pol[1, 1] with [h, u]=[4, 1] has ratio=1.3326166955388196603 E-114759 at X, ratio=2.6719307693795337093 E-101686 at Y, witness=2. Pol[2, 1] with [h, u]=[6, 2] has ratio=1.0266956398373537968 E-13069 at X, ratio=1.7937756553937938483 E-50848 at Y, witness=2. Pol[1, 2] with [h, u]=[4, 1] has ratio=4.442055651796065534 E-114760 at X, ratio=2.6719307693795337093 E-101686 at Y, witness=2. Pol[2, 2] with [h, u]=[6, 2] has ratio=1.0266956398373537968 E-13069 at X, ratio=5.381326966181381545 E-50848 at Y, witness=2. Validated in 4 sec. The certificate is available at here.

#### 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.
fieldvalue
prime_id135387
person_id9
machineUsing: Dual Intel Xeon Gold 5222 CPUs 3.8GHz
whatprp
notesCommand: /home/caldwell/clientpool/1/pfgw64 -t -q"2*7^709976+2*7^211441+1" 2>&1
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]
Primality testing 2*7^709976+2*7^211441+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 29.78%

2*7^709976+2*7^211441+1 is PRP! (3891.1029s+0.0079s)
[Elapsed time: 64.85 minutes]
modified2023-02-18 20:37:08
created2023-02-18 19:32:17
id181167

Query times: 0.0002 seconds to select prime, 0.0003 seconds to seek comments.