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Block #851,710

TWNLength 12β˜…β˜…β˜…β˜…β˜†

Bi-Twin Chain Β· Discovered 12/13/2014, 12:29:30 PM Β· Difficulty 10.9704 Β· 5,990,754 confirmations

TWN
Bi-Twin Chain

Interleaved pairs of primes that differ by 2, forming twin prime pairs at each level.

Block Header
Block Hash
0d8ce6ed8ec025bce630f6fd04216dbd15c9eaceeefc7be04c01dde51083d6c0
View on Chainz β†—

Height

#851,710

Difficulty

10.970439

Transactions

2

Size

399 B

Version

2

Bits

0af86ea9

Nonce

1,033,361,407

Timestamp

12/13/2014, 12:29:30 PM

Confirmations

5,990,754

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

710e34daa938c8e41f90b70f21a05c8dbc92573e1cb4badc51bd70af647278be
Transactions (2)
Coinbase729bbeea1c0a00…0347c8eff27390
1 in β†’ 1 out8.3220 XPM116 B
ANSXnLY2…Sd25TjkD
165909881dd7ed…a925b636a195b4
1 in β†’ 1 out0.0107 XPM192 B
AXoRvYm7…qc5rN7V9
Prime Chain Origin

This is the prime chain origin stored in the block header. It is a composite number (not prime itself) β€” it equals the first prime in the chain multiplied by a primorial. The origin anchors the entire chain to this specific block.

5.073 Γ— 10⁹⁷(98-digit number)
50737715594503047157…69923688168908763519
Discovered Prime Numbers
Lower: 2^k Γ— origin βˆ’ 1 | Upper: 2^k Γ— origin + 1

These are the actual prime numbers discovered by this block, computed using the verified Primecoin formula. Each number has been independently confirmed to pass the Fermat primality test. Use the FactorDB links to verify any number independently.

Level 0 β€” Twin Prime Pair (origin Β± 1)
origin βˆ’ 1
5.073 Γ— 10⁹⁷(98-digit number)
50737715594503047157…69923688168908763519
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
5.073 Γ— 10⁹⁷(98-digit number)
50737715594503047157…69923688168908763521
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: origin + 1 βˆ’ origin βˆ’ 1 = 2 (twin primes βœ“)
Level 1 β€” Twin Prime Pair (2^1 Γ— origin Β± 1)
2^1 Γ— origin βˆ’ 1
1.014 Γ— 10⁹⁸(99-digit number)
10147543118900609431…39847376337817527039
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
1.014 Γ— 10⁹⁸(99-digit number)
10147543118900609431…39847376337817527041
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^1 Γ— origin + 1 βˆ’ 2^1 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 2 β€” Twin Prime Pair (2^2 Γ— origin Β± 1)
2^2 Γ— origin βˆ’ 1
2.029 Γ— 10⁹⁸(99-digit number)
20295086237801218863…79694752675635054079
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
2.029 Γ— 10⁹⁸(99-digit number)
20295086237801218863…79694752675635054081
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^2 Γ— origin + 1 βˆ’ 2^2 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 3 β€” Twin Prime Pair (2^3 Γ— origin Β± 1)
2^3 Γ— origin βˆ’ 1
4.059 Γ— 10⁹⁸(99-digit number)
40590172475602437726…59389505351270108159
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
4.059 Γ— 10⁹⁸(99-digit number)
40590172475602437726…59389505351270108161
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^3 Γ— origin + 1 βˆ’ 2^3 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 4 β€” Twin Prime Pair (2^4 Γ— origin Β± 1)
2^4 Γ— origin βˆ’ 1
8.118 Γ— 10⁹⁸(99-digit number)
81180344951204875452…18779010702540216319
Verify on FactorDB β†—Wolfram Alpha β†—
2^4 Γ— origin + 1
8.118 Γ— 10⁹⁸(99-digit number)
81180344951204875452…18779010702540216321
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^4 Γ— origin + 1 βˆ’ 2^4 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 5 β€” Twin Prime Pair (2^5 Γ— origin Β± 1)
2^5 Γ— origin βˆ’ 1
1.623 Γ— 10⁹⁹(100-digit number)
16236068990240975090…37558021405080432639
Verify on FactorDB β†—Wolfram Alpha β†—
2^5 Γ— origin + 1
1.623 Γ— 10⁹⁹(100-digit number)
16236068990240975090…37558021405080432641
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^5 Γ— origin + 1 βˆ’ 2^5 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)

What this block proved

The miner who found this block proved the existence of 12 consecutive prime numbers forming a Bi-Twin Chain. The prime chain origin β€” the large number shown above β€” anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

β˜…β˜…β˜…β˜…β˜†
Rarity
ExceptionalChain length 12

Around 1 in 10,000 blocks. A significant mathematical achievement.

How Primecoin's Proof-of-Work Constructs These Primes

Primecoin stores a value called the prime chain origin in each block. The miner found a large integer such that when divided by a primorial (the product of small primes: 2 Γ— 3 Γ— 5 Γ— 7 Γ— …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial β€” that divisibility is part of the proof.

Prime Chain Origin = First Prime Γ— Primorial (2Β·3Β·5Β·7Β·11·…)
Source: Primecoin prime.cpp β€” CheckPrimeProofOfWork()

This is why the origin has many small prime factors β€” those factors are the primorial divisor. The chain then extends from the first prime using the TWN formula:

TWN: twin pairs (p, p+2) where p = origin/primorial βˆ’ 1 and p+2 = origin/primorial + 1
Cross-reference on Chainz Explorer
Circulating Supply:57,984,130 XPMΒ·at block #6,842,463 Β· updates every 60s
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