Block #301,727

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/9/2013, 9:48:36 AM · Difficulty 9.9925 · 6,493,823 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f0e71deff2f399da2d26d8ffe056f813de7591bc7f8470a43be1a3e7c9430846

View on Chainz ↗

Height

#301,727

Difficulty

9.992527

Transactions

Size

3.24 KB

Version

Bits

09fe1638

Nonce

258,595

Timestamp

12/9/2013, 9:48:36 AM

Confirmations

6,493,823

Mined by

⛏️ aRo|Ad9pSzHtZ9ebPysWxo4VY8quTmcpJ3y2zz

Merkle Root

711844729959d49d99a60b6e6fa4374415a8214f392bc79be78efbda81c58ff2

Transactions (4)

Coinbasea632fdfd04b526…98dfccf2493a7b

1 in → 31 out9.9800 XPM1.09 KB

Ad9pSzHt…cpJ3y2zz ARcqMJhp…RVLToQ6S AMbCuLHZ…WSoStwkc ASgn6o1Q…Ub9jXZvr AHjeGKcH…bHTM7J43

ce3ba5df864d89…892d8b6d7fb341

4 in → 2 out137.5204 XPM668 B

AMu1nqwY…bwxLqmNs ARH6oBCV…RVuRE922

114f8dc9d25fcf…287aa3528aeffb

6 in → 8 out22.3457 XPM1.08 KB

AKUmd8UD…5RbqPUAS AK5JvF4v…zoJguhWj AMMN5sWR…z8JYjFhB AHrxehtC…HcuNm3jU AQd7NAqj…KQ6cebzk

40ada1cf9eb076…1e8ed2349b45b9

2 in → 1 out6.0460 XPM340 B

AKtVDUTu…rQNNKfVW

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.

2.026 × 10⁹¹(92-digit number)

20263886392835559390…13361537365046929279

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

2.026 × 10⁹¹(92-digit number)

20263886392835559390…13361537365046929279

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.026 × 10⁹¹(92-digit number)

20263886392835559390…13361537365046929281

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

4.052 × 10⁹¹(92-digit number)

40527772785671118781…26723074730093858559

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.052 × 10⁹¹(92-digit number)

40527772785671118781…26723074730093858561

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

8.105 × 10⁹¹(92-digit number)

81055545571342237562…53446149460187717119

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.105 × 10⁹¹(92-digit number)

81055545571342237562…53446149460187717121

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

1.621 × 10⁹²(93-digit number)

16211109114268447512…06892298920375434239

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.621 × 10⁹²(93-digit number)

16211109114268447512…06892298920375434241

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

3.242 × 10⁹²(93-digit number)

32422218228536895025…13784597840750868479

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.242 × 10⁹²(93-digit number)

32422218228536895025…13784597840750868481

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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