Block #311,577

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 3:27:45 PM · Difficulty 9.9955 · 6,515,652 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

8a11b632b194290d933062aab4126ae8994bfe650e9eae87d5d634b2cac86fa1

View on Chainz ↗

Height

#311,577

Difficulty

9.995518

Transactions

Size

1.15 KB

Version

Bits

09feda4b

Nonce

35,608

Timestamp

12/14/2013, 3:27:45 PM

Confirmations

6,515,652

Mined by

⛏️ aRy7Aac9Hjdgnw4T4L2csqLecJsmZjP4ktypc3

Merkle Root

cac86e137bf3bd35fdd2c3157ff6e018ebdbbc66637d1adc8b31224fcee82d35

Transactions (1)

Coinbasecac86e137bf3bd…31224fcee82d35

1 in → 30 out9.9700 XPM1.06 KB

Aac9Hjdg…P4ktypc3 AZ2pPuKg…95SN2Yr7 Acskt6D1…Db4ci1L8 AG5YAjec…VhA4Aeh1 AdwTEXyC…NwbVbqZV

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.

5.530 × 10⁹³(94-digit number)

55308348911085607592…51957255625566127999

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

5.530 × 10⁹³(94-digit number)

55308348911085607592…51957255625566127999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.530 × 10⁹³(94-digit number)

55308348911085607592…51957255625566128001

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

1.106 × 10⁹⁴(95-digit number)

11061669782217121518…03914511251132255999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.106 × 10⁹⁴(95-digit number)

11061669782217121518…03914511251132256001

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

2.212 × 10⁹⁴(95-digit number)

22123339564434243037…07829022502264511999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.212 × 10⁹⁴(95-digit number)

22123339564434243037…07829022502264512001

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

4.424 × 10⁹⁴(95-digit number)

44246679128868486074…15658045004529023999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.424 × 10⁹⁴(95-digit number)

44246679128868486074…15658045004529024001

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

8.849 × 10⁹⁴(95-digit number)

88493358257736972148…31316090009058047999

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 #311,577

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