Block #221,051

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/21/2013, 8:54:31 AM · Difficulty 9.9379 · 6,589,414 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

6b3ebf64e5a04a7a6436705829f6726e706df5feeae2c9a8fbc9aadf56394f89

View on Chainz ↗

Height

#221,051

Difficulty

9.937900

Transactions

Size

1.97 KB

Version

Bits

09f01a37

Nonce

18,484

Timestamp

10/21/2013, 8:54:31 AM

Confirmations

6,589,414

Mined by

⛏️ P2SHAGeBQ3vDadzvrV9P8Mh9TTr9RdgVF3zZZP

Merkle Root

6b3fd4e3a416308f2a215a511cf97b918b07dba7502df77ed303283ec96b2add

Transactions (4)

Coinbase1b26df2d404f74…b44ea3ebf705f7

1 in → 1 out10.1400 XPM109 B

AGeBQ3vD…gVF3zZZP

b245f84bc9eea2…0398a7b2ba09df

3 in → 7 out20.3292 XPM623 B

AHhZFYzS…FdVBMtx9 AbuU1HhS…JPwsJEbB AL4L4dDV…KQBUViAD APLUhMH4…A9uUzFTE ARR7aRH6…ne3DXGXZ

9f90391ad1bcd3…b6e9edc7c5a4e5

6 in → 2 out15.0802 XPM967 B

AHnnkdch…FZepUEh7 AbbHSC6s…xe4cXGsS

f6bc62c201f0fa…6f67d7d0233f2c

1 in → 2 out8.0649 XPM225 B

ATps1pmh…yFVZin1Q AZScJLrw…CpdqTxKC

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.

1.604 × 10⁹⁵(96-digit number)

16042934639646851877…55797138042824762879

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

1.604 × 10⁹⁵(96-digit number)

16042934639646851877…55797138042824762879

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.604 × 10⁹⁵(96-digit number)

16042934639646851877…55797138042824762881

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

3.208 × 10⁹⁵(96-digit number)

32085869279293703755…11594276085649525759

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.208 × 10⁹⁵(96-digit number)

32085869279293703755…11594276085649525761

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

6.417 × 10⁹⁵(96-digit number)

64171738558587407510…23188552171299051519

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.417 × 10⁹⁵(96-digit number)

64171738558587407510…23188552171299051521

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

1.283 × 10⁹⁶(97-digit number)

12834347711717481502…46377104342598103039

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.283 × 10⁹⁶(97-digit number)

12834347711717481502…46377104342598103041

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

2.566 × 10⁹⁶(97-digit number)

25668695423434963004…92754208685196206079

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 #221,051

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