Block #211,223

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/15/2013, 2:40:13 PM · Difficulty 9.9152 · 6,585,223 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

45d0847d269eb0654278d007df2f231e57ea4b19d626f75e573a8b158df29f57

View on Chainz ↗

Height

#211,223

Difficulty

9.915237

Transactions

Size

4.86 KB

Version

Bits

09ea4cf6

Nonce

1,164,784,396

Timestamp

10/15/2013, 2:40:13 PM

Confirmations

6,585,223

Mined by

⛏️ 9RAZofRwGtUbNtymNhVyhAHtf285c53X46uM

Merkle Root

e3266ea2713c597c99417dc480805f2e2577570d2b6c84a11d490b0e11b83dbd

Transactions (2)

Coinbase7610ce0bf1d5a6…5a9a00a88a3e59

1 in → 122 out10.1100 XPM4.11 KB

AZofRwGt…c53X46uM ARx46AQ2…TwbEgb8H AdDjcFF6…qGKzhryp AMV2CWW7…yebsFBrS AckGXHge…bkNJhGsA

4d043e278932e8…d6f75cee131a6c

4 in → 2 out429.7809 XPM669 B

AVuTbupy…HXS73hdd Af3ZDREs…4mGdqDCP

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.

7.001 × 10¹⁰²(103-digit number)

70017662234818571096…77172700139614466559

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

7.001 × 10¹⁰²(103-digit number)

70017662234818571096…77172700139614466559

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.001 × 10¹⁰²(103-digit number)

70017662234818571096…77172700139614466561

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

1.400 × 10¹⁰³(104-digit number)

14003532446963714219…54345400279228933119

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.400 × 10¹⁰³(104-digit number)

14003532446963714219…54345400279228933121

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

2.800 × 10¹⁰³(104-digit number)

28007064893927428438…08690800558457866239

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.800 × 10¹⁰³(104-digit number)

28007064893927428438…08690800558457866241

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

5.601 × 10¹⁰³(104-digit number)

56014129787854856877…17381601116915732479

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.601 × 10¹⁰³(104-digit number)

56014129787854856877…17381601116915732481

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

1.120 × 10¹⁰⁴(105-digit number)

11202825957570971375…34763202233831464959

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