Block #321,844

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/20/2013, 1:51:58 PM · Difficulty 10.1982 · 6,503,871 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

802736b0520837ff4fc953c2137ed01a3a9c2b5915c3c4dba30245be2281a962

View on Chainz ↗

Height

#321,844

Difficulty

10.198175

Transactions

Size

3.41 KB

Version

Bits

0a32bb9a

Nonce

85,430

Timestamp

12/20/2013, 1:51:58 PM

Confirmations

6,503,871

Mined by

⛏️ 4aRKcxAG5YAjechu8kNZ5eyyVJVz1YnBVhA4Aeh1

Merkle Root

3b0c09a15eeae6724d700f34597d15c65acf0c73abbc0183a6209940e76f60a0

Transactions (4)

Coinbase5382a0d016a62d…94569e756f7a42

1 in → 28 out9.5800 XPM1015 B

AG5YAjec…VhA4Aeh1 Aac9Hjdg…P4ktypc3 Ad9pSzHt…cpJ3y2zz AJYusoxy…LtcN2Chh APJPNQaM…8nyTxzkW

e0c0772bd581dc…8953f97d5ae7f2

1 in → 2 out14.9900 XPM225 B

AZvdYS4d…1aQBmYsH ASDbs3jg…XG5ij6Ea

f7416f4b4ce283…c7e84ea5772f03

2 in → 1 out9.8800 XPM339 B

AWQ7Md9r…eNV1Kva2

45a9c68fba52e0…fa97eb12d92cb1

12 in → 1 out3.7563 XPM1.78 KB

AWBivb58…iiSScBhf

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.

1.668 × 10⁹⁷(98-digit number)

16685815154352421561…79819936048285442559

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

1.668 × 10⁹⁷(98-digit number)

16685815154352421561…79819936048285442559

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.668 × 10⁹⁷(98-digit number)

16685815154352421561…79819936048285442561

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

3.337 × 10⁹⁷(98-digit number)

33371630308704843122…59639872096570885119

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.337 × 10⁹⁷(98-digit number)

33371630308704843122…59639872096570885121

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

6.674 × 10⁹⁷(98-digit number)

66743260617409686244…19279744193141770239

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.674 × 10⁹⁷(98-digit number)

66743260617409686244…19279744193141770241

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

1.334 × 10⁹⁸(99-digit number)

13348652123481937248…38559488386283540479

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.334 × 10⁹⁸(99-digit number)

13348652123481937248…38559488386283540481

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

2.669 × 10⁹⁸(99-digit number)

26697304246963874497…77118976772567080959

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.669 × 10⁹⁸(99-digit number)

26697304246963874497…77118976772567080961

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 #321,844

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