Block #321,419

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/20/2013, 7:38:34 AM · Difficulty 10.1897 · 6,496,407 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

721c334d41932c05fe754bec4d01e891a65d5cd55c04cb75157bef299f2b8cf8

View on Chainz ↗

Height

#321,419

Difficulty

10.189733

Transactions

Size

573 B

Version

Bits

0a30925f

Nonce

92,478

Timestamp

12/20/2013, 7:38:34 AM

Confirmations

6,496,407

Mined by

⛏️ P2SHAGMZAp6ed2B2GrdLz7V2fDtE36Kyh1yRYD

Merkle Root

a69d985ce694ba16ed71d7bf74d8fe7f71098777b9e9fc0a8e3b358d7c075769

Transactions (2)

Coinbase133d6605b0499b…d5ce69338d698a

1 in → 1 out9.6300 XPM109 B

AGMZAp6e…Kyh1yRYD

160ba11ac55d61…13ff600c8305d8

2 in → 2 out265.7897 XPM373 B

AbzCehaV…JNzyJ3DR AHTkdmKb…rQVQVrWi

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.

8.431 × 10⁹⁵(96-digit number)

84315081498664048339…44831928128661432319

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

8.431 × 10⁹⁵(96-digit number)

84315081498664048339…44831928128661432319

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.431 × 10⁹⁵(96-digit number)

84315081498664048339…44831928128661432321

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

1.686 × 10⁹⁶(97-digit number)

16863016299732809667…89663856257322864639

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.686 × 10⁹⁶(97-digit number)

16863016299732809667…89663856257322864641

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

3.372 × 10⁹⁶(97-digit number)

33726032599465619335…79327712514645729279

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.372 × 10⁹⁶(97-digit number)

33726032599465619335…79327712514645729281

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

6.745 × 10⁹⁶(97-digit number)

67452065198931238671…58655425029291458559

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.745 × 10⁹⁶(97-digit number)

67452065198931238671…58655425029291458561

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

1.349 × 10⁹⁷(98-digit number)

13490413039786247734…17310850058582917119

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.349 × 10⁹⁷(98-digit number)

13490413039786247734…17310850058582917121

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

Block #321,419

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