Block #13,731

TWNLength 7★☆☆☆☆

Bi-Twin Chain · Discovered 7/11/2013, 2:18:16 PM · Difficulty 7.8047 · 6,781,047 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

0c8510cafa68cf1d29ac410e1ed41368042cae2ad296d1d9c06f7a1d6b7ad6ce

View on Chainz ↗

Height

#13,731

Difficulty

7.804727

Transactions

Size

965 B

Version

Bits

07ce0293

Nonce

158

Timestamp

7/11/2013, 2:18:16 PM

Confirmations

6,781,047

Mined by

⛏️ P2SHATAPoK2KGpbq55AcQx7nhwhWq9NqN7EWoy

Merkle Root

7d47b5e7c99afaee5a9882b0bae602b18b250994b17a4789793b187dbd2959db

Transactions (3)

Coinbaseb51d90b2f78922…f6727ba8bca2bf

1 in → 1 out16.4200 XPM108 B

ATAPoK2K…NqN7EWoy

d1422a09ca458b…ffb18f65a57870

5 in → 1 out91.0300 XPM613 B

AYLXzB77…zfoMnfnQ

7382933bba5dca…64f601aa921e0d

1 in → 1 out17.1800 XPM158 B

ALkUzjQM…ittmb2bv

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.

1.058 × 10⁸⁶(87-digit number)

10580627602913509677…36276477011565173699

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

1.058 × 10⁸⁶(87-digit number)

10580627602913509677…36276477011565173699

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.058 × 10⁸⁶(87-digit number)

10580627602913509677…36276477011565173701

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

2.116 × 10⁸⁶(87-digit number)

21161255205827019355…72552954023130347399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.116 × 10⁸⁶(87-digit number)

21161255205827019355…72552954023130347401

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

4.232 × 10⁸⁶(87-digit number)

42322510411654038711…45105908046260694799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.232 × 10⁸⁶(87-digit number)

42322510411654038711…45105908046260694801

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

8.464 × 10⁸⁶(87-digit number)

84645020823308077422…90211816092521389599

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 7 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 7

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