Block #211,612

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/15/2013, 7:28:47 PM · Difficulty 9.9169 · 6,593,662 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

247255c23c82c930c7276a6545e477ba5a91a54f8c19ae2a68d0aaa2cf967790

View on Chainz ↗

Height

#211,612

Difficulty

9.916896

Transactions

Size

1.85 KB

Version

Bits

09eab9b4

Nonce

21,076

Timestamp

10/15/2013, 7:28:47 PM

Confirmations

6,593,662

Mined by

⛏️ P2SHAXkwZGZQUV996sgzkcs2jbamXUaP3jC9dV

Merkle Root

01a97ef652ddea3ddc8b5670e930549b435f1ac0c9614400674803661bc3229e

Transactions (4)

Coinbasef7e2aca367b66d…ef4afae47c30ac

1 in → 1 out10.1800 XPM109 B

AXkwZGZQ…aP3jC9dV

83a147842c702c…f9bc17f9581a55

5 in → 3 out1259.3774 XPM852 B

AZD5zWVk…WaHRqp5F AdxC9tpP…F5PhSark AYtMiX9E…Q2uYCGgD

71ad8a7959134c…8d0aa9539cd562

5 in → 1 out50.9700 XPM615 B

AbxSykSg…BX71twYi

b2b156f55922b3…c94a88ddb9bd7e

1 in → 2 out3.1299 XPM226 B

AdWp2YRv…FDtt217a AGqBA3Ts…QsargvL6

Prime Chain Origin

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

3.338 × 10⁹⁷(98-digit number)

33386308013485475499…46823477324467903999

Discovered Prime Numbers

Lower: 2^k × origin − 1 | Upper: 2^k × 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 — Twin Prime Pair (origin ± 1)

origin − 1

3.338 × 10⁹⁷(98-digit number)

33386308013485475499…46823477324467903999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.338 × 10⁹⁷(98-digit number)

33386308013485475499…46823477324467904001

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: origin + 1 − origin − 1 = 2 (twin primes ✓)

Level 1 — Twin Prime Pair (2^1 × origin ± 1)

2^1 × origin − 1

6.677 × 10⁹⁷(98-digit number)

66772616026970950999…93646954648935807999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.677 × 10⁹⁷(98-digit number)

66772616026970950999…93646954648935808001

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^1 × origin + 1 − 2^1 × origin − 1 = 2 (twin primes ✓)

Level 2 — Twin Prime Pair (2^2 × origin ± 1)

2^2 × origin − 1

1.335 × 10⁹⁸(99-digit number)

13354523205394190199…87293909297871615999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.335 × 10⁹⁸(99-digit number)

13354523205394190199…87293909297871616001

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^2 × origin + 1 − 2^2 × origin − 1 = 2 (twin primes ✓)

Level 3 — Twin Prime Pair (2^3 × origin ± 1)

2^3 × origin − 1

2.670 × 10⁹⁸(99-digit number)

26709046410788380399…74587818595743231999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.670 × 10⁹⁸(99-digit number)

26709046410788380399…74587818595743232001

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

Level 4 — Twin Prime Pair (2^4 × origin ± 1)

2^4 × origin − 1

5.341 × 10⁹⁸(99-digit number)

53418092821576760799…49175637191486463999

Verify on FactorDB ↗Wolfram Alpha ↗

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 — the large number shown above — anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

★☆☆☆☆

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 × 3 × 5 × 7 × …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial — that divisibility is part of the proof.

Prime Chain Origin = First Prime × Primorial (2·3·5·7·11·…)

Source: Primecoin prime.cpp — CheckPrimeProofOfWork()

This is why the origin has many small prime factors — 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 − 1 and p+2 = origin/primorial + 1

Cross-reference on Chainz Explorer