Block #213,010

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/16/2013, 3:41:05 PM · Difficulty 9.9199 · 6,612,121 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

57c01a97eef84e8778efd600673ee96716cfed381a32bdca90bec628a3d66ff0

View on Chainz ↗

Height

#213,010

Difficulty

9.919889

Transactions

Size

5.46 KB

Version

Bits

09eb7dd2

Nonce

1,164,749,889

Timestamp

10/16/2013, 3:41:05 PM

Confirmations

6,612,121

Mined by

⛏️ @R^{AViH25QJsh8wVdewnBan44qXA8kfkafa8f

Merkle Root

b0a0372c4ea145f6187eff4da9230bbb4e6722b77fcc6df4d5226fce9e7e7a1e

Transactions (1)

Coinbaseb0a0372c4ea145…226fce9e7e7a1e

1 in → 160 out10.0900 XPM5.37 KB

AViH25QJ…kfkafa8f ANfpVCC4…Qu8GmRLT AMttQDuf…7j6tfS9x AHFF7EtH…d8HHoKnf AbGUqMnw…abP9ZPcY

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.

4.347 × 10⁹⁷(98-digit number)

43472864974803188942…92521657116692367359

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

4.347 × 10⁹⁷(98-digit number)

43472864974803188942…92521657116692367359

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.347 × 10⁹⁷(98-digit number)

43472864974803188942…92521657116692367361

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

8.694 × 10⁹⁷(98-digit number)

86945729949606377884…85043314233384734719

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.694 × 10⁹⁷(98-digit number)

86945729949606377884…85043314233384734721

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

1.738 × 10⁹⁸(99-digit number)

17389145989921275576…70086628466769469439

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.738 × 10⁹⁸(99-digit number)

17389145989921275576…70086628466769469441

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

3.477 × 10⁹⁸(99-digit number)

34778291979842551153…40173256933538938879

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.477 × 10⁹⁸(99-digit number)

34778291979842551153…40173256933538938881

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

6.955 × 10⁹⁸(99-digit number)

69556583959685102307…80346513867077877759

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

Block #213,010

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