Block #265,612

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 11/19/2013, 5:44:31 PM · Difficulty 9.9621 · 6,540,420 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

81bfc8a8b327d5c4b952a864e41faf3d0630decdd4831669b6dd316d678f3fe6

View on Chainz ↗

Height

#265,612

Difficulty

9.962131

Transactions

Size

2.83 KB

Version

Bits

09f64e3c

Nonce

5,013

Timestamp

11/19/2013, 5:44:31 PM

Confirmations

6,540,420

Mined by

⛏️ aRT\AP2GxFDi7THJ5XysifAM1TroxMMZQBdSYt

Merkle Root

08a400c4404778214c1e60f98be03b45ff15d8c4da4c3528b9a0b3a85d853137

Transactions (4)

Coinbase7e0b8e8bcc147b…8dbff654d5c123

1 in → 52 out10.0400 XPM1.79 KB

AP2GxFDi…MZQBdSYt ATXX3EoZ…FWgsrz7q APZy8y6E…QZaGQjfp ANHbaxZp…XjnHCJJs AVzhvfTX…ZzNJYq61

261f54167f4e0a…d8b40653cb1674

2 in → 2 out20.0500 XPM374 B

AYSi8YAE…WzANgkBX AP7anwzL…RBSGSGEP

92b967afd007f1…230fa499a40e80

2 in → 2 out20.0500 XPM374 B

AcFcH9pe…xaMrL5VC Aa2JA2Lk…R4Q3S5WK

43984823bd9d83…b79b2d74a24651

1 in → 2 out10.0200 XPM226 B

AN6bAFEn…Pxsc8CgB AJVF4WMT…d5ew7aSC

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.

1.129 × 10⁹⁴(95-digit number)

11294559244455041931…04095621289083422719

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

1.129 × 10⁹⁴(95-digit number)

11294559244455041931…04095621289083422719

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.129 × 10⁹⁴(95-digit number)

11294559244455041931…04095621289083422721

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

2.258 × 10⁹⁴(95-digit number)

22589118488910083863…08191242578166845439

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.258 × 10⁹⁴(95-digit number)

22589118488910083863…08191242578166845441

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

4.517 × 10⁹⁴(95-digit number)

45178236977820167727…16382485156333690879

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.517 × 10⁹⁴(95-digit number)

45178236977820167727…16382485156333690881

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

9.035 × 10⁹⁴(95-digit number)

90356473955640335455…32764970312667381759

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

9.035 × 10⁹⁴(95-digit number)

90356473955640335455…32764970312667381761

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

1.807 × 10⁹⁵(96-digit number)

18071294791128067091…65529940625334763519

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.807 × 10⁹⁵(96-digit number)

18071294791128067091…65529940625334763521

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