3⁸⁵⁰⁵⁰⁰ + 3²⁴⁸⁹¹⁸ - 1

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Lei Zhou writes (16 Jul 2016):  (report abuse)

This is a balanced ternary prime with only 3 non-zero digits (in balanced ternary base). p+1 = 3^850500 + 3^248918 = Phi(4,3)*Phi(244,3)*Phi(19724,3)*Phi(1203164,3) which has the following small factors: 2, 3^248918, 5, 98621, 180317, 1518749, 1538473, 262199473, 14033606981014657, 39504363995133913, 125904201746877738245401, 865923475887669700104067517 Using the above small factors as help, OpenPFGW provides 87.89% proof: Primality testing 3^850500+3^248918-1 [N-1/N+1, Brillhart-Lehmer-Selfridge] Primality testing 3^850500+3^248918-1 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 3 Running N+1 test using discriminant 17, base 8+sqrt(17) Calling N+1 BLS with factored part 29.30% and helper 0.00% (87.89% proof) 3^850500+3^248918-1 is Fermat and Lucas PRP! (22180.7503s+0.0095s) Using F = 2 and G = the product of the above listed small factors, 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_850500_+3_248918_-1.in". Pol[1, 1] with [h, u]=[4, 1] has ratio=6.850021549422434505 E-69380 at X, ratio=9.193825434597816829 E-81355 at Y, witness=2. Pol[2, 1] with [h, u]=[5, 2] has ratio=1.0279009322513632653 E-81354 at X, ratio=1.0138544926424912111 E-40677 at Y, witness=2. Pol[3, 1] with [h, u]=[6, 2] has ratio=3.046034677385834765 E-28588 at X, ratio=3.578658306755167131 E-20454 at Y, witness=5. Validated in 6 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.