Block #264,191

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/18/2013, 11:25:28 AM · Difficulty 9.9649 · 6,550,021 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

5419567c1da48ad8c5c064d8455193ced5f938f32b4fcb5300c84cf440f46a02

View on Chainz ↗

Height

#264,191

Difficulty

9.964939

Transactions

Size

650 B

Version

Bits

09f70646

Nonce

5,461

Timestamp

11/18/2013, 11:25:28 AM

Confirmations

6,550,021

Mined by

⛏️ P2SHAdGRRh1eP4JFvQMqy7y2pLsryDQmctLb24

Merkle Root

81a1be0c182c17a6309b801f9779342ec77d56c07511133b9aaac303427ce6c6

Transactions (3)

Coinbasea689046b7208e6…9bc8a1fdfef43c

1 in → 2 out10.0800 XPM141 B

AdGRRh1e…QmctLb24 ASNt3cJa…L5zCqAYM

482f4000ff9036…002f1a084d45b0

1 in → 3 out10.0300 XPM226 B

AcADmAoe…8iNnoMzB AeKNrjNj…1NEq95Ta AMuPGPXT…eMMNvd8v

6a827cc465fe91…8e3d761d157842

1 in → 1 out174.0558 XPM193 B

ATknEwpv…tz6xvy1e

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.

8.353 × 10⁹⁵(96-digit number)

83532392329605390305…28924010179546839759

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

8.353 × 10⁹⁵(96-digit number)

83532392329605390305…28924010179546839759

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.353 × 10⁹⁵(96-digit number)

83532392329605390305…28924010179546839761

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

1.670 × 10⁹⁶(97-digit number)

16706478465921078061…57848020359093679519

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.670 × 10⁹⁶(97-digit number)

16706478465921078061…57848020359093679521

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

3.341 × 10⁹⁶(97-digit number)

33412956931842156122…15696040718187359039

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.341 × 10⁹⁶(97-digit number)

33412956931842156122…15696040718187359041

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

6.682 × 10⁹⁶(97-digit number)

66825913863684312244…31392081436374718079

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.682 × 10⁹⁶(97-digit number)

66825913863684312244…31392081436374718081

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

1.336 × 10⁹⁷(98-digit number)

13365182772736862448…62784162872749436159

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 #264,191

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