Block #214,234

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/17/2013, 8:51:29 AM · Difficulty 9.9230 · 6,595,618 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

8fc21ef3137c28380f09218961c7e6f2c2c97759cfb703d4682d4b638d821d9c

View on Chainz ↗

Height

#214,234

Difficulty

9.922987

Transactions

Size

1.00 KB

Version

Bits

09ec48d8

Nonce

247,183

Timestamp

10/17/2013, 8:51:29 AM

Confirmations

6,595,618

Mined by

⛏️ P2SHAHnC2vHTnpRWffuNbHGv3Mh3M9SjbyyVBp

Merkle Root

eab9e56fda1b1bb9c9f7aa45044e0ce0111eaeedef9133b18710c8aa7c827e5a

Transactions (4)

Coinbase4c3feefa9945b3…5a6b2b069d244c

1 in → 1 out10.1700 XPM109 B

AHnC2vHT…SjbyyVBp

59452d5bb06ae2…3cbb2c932a1819

1 in → 2 out8.1322 XPM225 B

AY355AXA…eZ7GEPo5 Ab3dSvM7…Qoxxb9tp

9ccb3f11f53030…6a8c03458d2a4a

1 in → 2 out6.0396 XPM226 B

AY4uM53e…yee8uDKH APMggGfg…cUPa7zb7

ad0c69acfbb68f…082c44d9eddf56

2 in → 2 out2.2909 XPM373 B

AcJ4TARz…FMfqvTQR Abx5qH6G…Hh56BvNt

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.

2.238 × 10⁹⁷(98-digit number)

22386719232545144722…84857893154461025919

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

2.238 × 10⁹⁷(98-digit number)

22386719232545144722…84857893154461025919

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.238 × 10⁹⁷(98-digit number)

22386719232545144722…84857893154461025921

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

4.477 × 10⁹⁷(98-digit number)

44773438465090289444…69715786308922051839

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.477 × 10⁹⁷(98-digit number)

44773438465090289444…69715786308922051841

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

8.954 × 10⁹⁷(98-digit number)

89546876930180578888…39431572617844103679

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.954 × 10⁹⁷(98-digit number)

89546876930180578888…39431572617844103681

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

1.790 × 10⁹⁸(99-digit number)

17909375386036115777…78863145235688207359

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.790 × 10⁹⁸(99-digit number)

17909375386036115777…78863145235688207361

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

3.581 × 10⁹⁸(99-digit number)

35818750772072231555…57726290471376414719

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 #214,234

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