Block #310,813

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 6:25:04 AM · Difficulty 9.9953 · 6,504,328 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e4a8c5f48e8619c76f72bab85110f977365331ba9756f1d5435311cdb10315dc

View on Chainz ↗

Height

#310,813

Difficulty

9.995292

Transactions

Size

1.21 KB

Version

Bits

09fecb70

Nonce

16,187

Timestamp

12/14/2013, 6:25:04 AM

Confirmations

6,504,328

Mined by

⛏️ aR'AZ3bZpffwRyTH6paPfbEovHQkgLnRQBGfh

Merkle Root

c7c46e635609d8a588d009c2c61bdbc6142b9f9a85102c79b71ac370ea31cfc0

Transactions (4)

Coinbase1af3d43a93babd…d8fec581c0d94c

1 in → 12 out9.9900 XPM471 B

AZ3bZpff…LnRQBGfh AJzWx2VD…j4ZSMit5 AMdvB6BS…DPq9FYdA AbtgNzNb…9Atvu81w AKwqFUdz…D4jGNKFN

7cf8d7b7e422fc…fb9d2c65628388

1 in → 2 out7.9407 XPM227 B

AcLDvygp…GAR414Po AP4DKnjy…BeqMS5Mp

38b036f2f69d64…60076fffe53c36

1 in → 2 out6.9135 XPM225 B

ATdV4gn5…qCyGeGw1 AJFjtYKD…Z4WHivy1

78d7edb1e96d37…38c166251916f4

1 in → 2 out6.8368 XPM226 B

ANGjw2xm…Y7UyhDH7 AVZfTMTo…cqZv9wmp

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.

1.431 × 10⁹⁹(100-digit number)

14311357634111166171…10484454866925808639

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

1.431 × 10⁹⁹(100-digit number)

14311357634111166171…10484454866925808639

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.431 × 10⁹⁹(100-digit number)

14311357634111166171…10484454866925808641

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

2.862 × 10⁹⁹(100-digit number)

28622715268222332342…20968909733851617279

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.862 × 10⁹⁹(100-digit number)

28622715268222332342…20968909733851617281

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

5.724 × 10⁹⁹(100-digit number)

57245430536444664684…41937819467703234559

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.724 × 10⁹⁹(100-digit number)

57245430536444664684…41937819467703234561

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

1.144 × 10¹⁰⁰(101-digit number)

11449086107288932936…83875638935406469119

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.144 × 10¹⁰⁰(101-digit number)

11449086107288932936…83875638935406469121

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

2.289 × 10¹⁰⁰(101-digit number)

22898172214577865873…67751277870812938239

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 #310,813

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