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Block #267,337

TWNLength 9β˜…β˜†β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 11/21/2013, 5:08:33 AM Β· Difficulty 9.9591 Β· 6,527,690 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
a29aa52d41b6a6d029350e9c4fa17d0b3a8bdec37448725328d2e9eefe1c1914
View on Chainz β†—

Height

#267,337

Difficulty

9.959090

Transactions

1

Size

211 B

Version

2

Bits

09f586ef

Nonce

186

Timestamp

11/21/2013, 5:08:33 AM

Confirmations

6,527,690

Mined by

⛏️ P2SHAcLBm5Z9BD7o7btAchFh9KZEkSS683d7c3

Merkle Root

581c1b6cc0b9f1158c81663a0c3c0ecb78765e2453b8e0cd7f2cd196f51fffc6
Transactions (1)
Coinbase581c1b6cc0b9f1…2cd196f51fffc6
1 in β†’ 1 out10.0700 XPM116 B
AcLBm5Z9…S683d7c3
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.

2.285 Γ— 10¹⁰⁷(108-digit number)
22858453087269196036…00095328289054719999
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
2.285 Γ— 10¹⁰⁷(108-digit number)
22858453087269196036…00095328289054719999
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
2.285 Γ— 10¹⁰⁷(108-digit number)
22858453087269196036…00095328289054720001
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
4.571 Γ— 10¹⁰⁷(108-digit number)
45716906174538392072…00190656578109439999
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
4.571 Γ— 10¹⁰⁷(108-digit number)
45716906174538392072…00190656578109440001
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
9.143 Γ— 10¹⁰⁷(108-digit number)
91433812349076784144…00381313156218879999
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
9.143 Γ— 10¹⁰⁷(108-digit number)
91433812349076784144…00381313156218880001
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
1.828 Γ— 10¹⁰⁸(109-digit number)
18286762469815356828…00762626312437759999
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
1.828 Γ— 10¹⁰⁸(109-digit number)
18286762469815356828…00762626312437760001
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
3.657 Γ— 10¹⁰⁸(109-digit number)
36573524939630713657…01525252624875519999
Verify on FactorDB β†—Wolfram Alpha β†—

What this block proved

The miner who found this block proved the existence of 9 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
CommonChain length 9

Found in most blocks. The baseline for Primecoin's proof-of-work.

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:57,604,263 XPMΒ·at block #6,795,026 Β· updates every 60s
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