Block #51,319

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/16/2013, 5:16:29 AM · Difficulty 8.8966 · 6,743,460 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

baae3cd37b5da8dadc94b19546d9b990833fa38262de1d00e9008dc8f01ae03f

View on Chainz ↗

Height

#51,319

Difficulty

8.896647

Transactions

Size

511 B

Version

Bits

08e58aa3

Nonce

Timestamp

7/16/2013, 5:16:29 AM

Confirmations

6,743,460

Mined by

⛏️ P2SHAWXuVKXDmHLwiZFnquW96aKA9TgFeUL2wY

Merkle Root

109530c31fe12c6effd2c239ae1ab9a22b96cbde4d53381f47111c2cd48b328c

Transactions (2)

Coinbase0204f8b86a7541…6dc2ed9e075a03

1 in → 1 out12.6300 XPM110 B

AWXuVKXD…gFeUL2wY

f7dfe0c67d1c21…4bdd8296a3a511

2 in → 1 out12.9100 XPM306 B

AbjLQB3Y…SXLCXwVY

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.

1.535 × 10¹⁰⁶(107-digit number)

15353912746083571451…54408238155795047389

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

1.535 × 10¹⁰⁶(107-digit number)

15353912746083571451…54408238155795047389

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.535 × 10¹⁰⁶(107-digit number)

15353912746083571451…54408238155795047391

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

3.070 × 10¹⁰⁶(107-digit number)

30707825492167142902…08816476311590094779

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.070 × 10¹⁰⁶(107-digit number)

30707825492167142902…08816476311590094781

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

6.141 × 10¹⁰⁶(107-digit number)

61415650984334285805…17632952623180189559

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.141 × 10¹⁰⁶(107-digit number)

61415650984334285805…17632952623180189561

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

1.228 × 10¹⁰⁷(108-digit number)

12283130196866857161…35265905246360379119

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.228 × 10¹⁰⁷(108-digit number)

12283130196866857161…35265905246360379121

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 8 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 8

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