Block #213,830

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/17/2013, 2:28:26 AM · Difficulty 9.9226 · 6,593,876 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

83933bbdf89c4b4fa6688547a166d207613e8aec5ee5139893d8085a1c8373c9

View on Chainz ↗

Height

#213,830

Difficulty

9.922635

Transactions

Size

4.26 KB

Version

Bits

09ec31d3

Nonce

1,164,775,605

Timestamp

10/17/2013, 2:28:26 AM

Confirmations

6,593,876

Mined by

⛏️ FCR_K7AP5d2TUfDS1ccjgrv2MvPBrWTwP2wUZ8fL

Merkle Root

e7194cd220bdea5dce93037d0369de8d5b58771aa553af258bc416139ddaa9c8

Transactions (1)

Coinbasee7194cd220bdea…c416139ddaa9c8

1 in → 124 out10.0900 XPM4.18 KB

AP5d2TUf…P2wUZ8fL AcV2etJ3…AC6C3Zed AHwj9V9X…eJXSvGKX AWAQowiv…NhXy6813 AP9uYfvU…WieGQKa8

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.

1.469 × 10⁹⁴(95-digit number)

14690392389254227686…83183506369823247359

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

1.469 × 10⁹⁴(95-digit number)

14690392389254227686…83183506369823247359

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.469 × 10⁹⁴(95-digit number)

14690392389254227686…83183506369823247361

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

2.938 × 10⁹⁴(95-digit number)

29380784778508455372…66367012739646494719

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.938 × 10⁹⁴(95-digit number)

29380784778508455372…66367012739646494721

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

5.876 × 10⁹⁴(95-digit number)

58761569557016910745…32734025479292989439

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.876 × 10⁹⁴(95-digit number)

58761569557016910745…32734025479292989441

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

1.175 × 10⁹⁵(96-digit number)

11752313911403382149…65468050958585978879

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.175 × 10⁹⁵(96-digit number)

11752313911403382149…65468050958585978881

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

2.350 × 10⁹⁵(96-digit number)

23504627822806764298…30936101917171957759

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 #213,830

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