Block #215,710

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 10/18/2013, 6:47:41 AM · Difficulty 9.9254 · 6,589,434 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e031f8ed505992655c8fe07f9889fed4ba8bdda99cad68340d7b88ec4031a401

View on Chainz ↗

Height

#215,710

Difficulty

9.925447

Transactions

Size

5.46 KB

Version

Bits

09ecea15

Nonce

78,193

Timestamp

10/18/2013, 6:47:41 AM

Confirmations

6,589,434

Mined by

⛏️ JR`gAP5d2TUfDS1ccjgrv2MvPBrWTwP2wUZ8fL

Merkle Root

184132baccc5106d27092e1bc1d6a0d104af5c03b51fa3ba65067851e0071735

Transactions (1)

Coinbase184132baccc510…067851e0071735

1 in → 160 out10.0800 XPM5.37 KB

AP5d2TUf…P2wUZ8fL AHKLJG5f…VFkk9jqW ALhiG8Wa…tcZGAG9C AM4zvBDn…dBXWgwDw AR3yVDg3…SFPbbCst

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.

2.026 × 10⁹⁶(97-digit number)

20267987250157011617…90190446407069800799

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

2.026 × 10⁹⁶(97-digit number)

20267987250157011617…90190446407069800799

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.026 × 10⁹⁶(97-digit number)

20267987250157011617…90190446407069800801

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

4.053 × 10⁹⁶(97-digit number)

40535974500314023235…80380892814139601599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.053 × 10⁹⁶(97-digit number)

40535974500314023235…80380892814139601601

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

8.107 × 10⁹⁶(97-digit number)

81071949000628046471…60761785628279203199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.107 × 10⁹⁶(97-digit number)

81071949000628046471…60761785628279203201

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

1.621 × 10⁹⁷(98-digit number)

16214389800125609294…21523571256558406399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.621 × 10⁹⁷(98-digit number)

16214389800125609294…21523571256558406401

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

3.242 × 10⁹⁷(98-digit number)

32428779600251218588…43047142513116812799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.242 × 10⁹⁷(98-digit number)

32428779600251218588…43047142513116812801

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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