Block #213,640

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/16/2013, 11:52:25 PM · Difficulty 9.9221 · 6,600,646 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

c0c0e4a7f079b1da8100514c1fd63d3a44dd4301e9c1be0e583ea2bdc9f25c80

View on Chainz ↗

Height

#213,640

Difficulty

9.922085

Transactions

Size

5.53 KB

Version

Bits

09ec0dc3

Nonce

1,164,734,458

Timestamp

10/16/2013, 11:52:25 PM

Confirmations

6,600,646

Mined by

⛏️ BR_&dAPw6W8tkxffFh3841R9HsV9zdPRgVVbThY

Merkle Root

ccbf805d19c2dfbde9140aa1c0d3667c6065a00baa48100a2c54fbbf3440aad1

Transactions (1)

Coinbaseccbf805d19c2df…54fbbf3440aad1

1 in → 162 out10.0800 XPM5.44 KB

APw6W8tk…RgVVbThY AJWc12T9…gdZnUvYf AauXrWxt…MwUoygQt AHMctDwj…kqtWUS3H ALNG6kUM…K1kfhET3

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.

2.976 × 10⁹⁹(100-digit number)

29767044946407412372…50835896089138665599

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

2.976 × 10⁹⁹(100-digit number)

29767044946407412372…50835896089138665599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.976 × 10⁹⁹(100-digit number)

29767044946407412372…50835896089138665601

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

5.953 × 10⁹⁹(100-digit number)

59534089892814824745…01671792178277331199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.953 × 10⁹⁹(100-digit number)

59534089892814824745…01671792178277331201

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

1.190 × 10¹⁰⁰(101-digit number)

11906817978562964949…03343584356554662399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.190 × 10¹⁰⁰(101-digit number)

11906817978562964949…03343584356554662401

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

2.381 × 10¹⁰⁰(101-digit number)

23813635957125929898…06687168713109324799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.381 × 10¹⁰⁰(101-digit number)

23813635957125929898…06687168713109324801

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

4.762 × 10¹⁰⁰(101-digit number)

47627271914251859796…13374337426218649599

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 #213,640

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