Block #57,354

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/17/2013, 1:57:47 PM · Difficulty 8.9541 · 6,744,280 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e53ba1b9e9e0860d68909af3971f4b17e5023cd70397122b86e596c8e9e15a0c

View on Chainz ↗

Height

#57,354

Difficulty

8.954072

Transactions

Size

2.17 KB

Version

Bits

08f43e12

Nonce

259

Timestamp

7/17/2013, 1:57:47 PM

Confirmations

6,744,280

Mined by

⛏️ P2SHAWzWopsJHNp68y6kxspePhf9pPiocD7Jca

Merkle Root

0d4420005a8286e863a3edeb3a48eea8f8c0ddf0df5aec710a85473dc085347b

Transactions (2)

Coinbase821c35bd70f8b4…de4da97377d835

1 in → 1 out12.4900 XPM110 B

AWzWopsJ…iocD7Jca

959a9d768cd67e…7a187dd1065692

17 in → 2 out213.7700 XPM1.97 KB

AJbRF9Vh…cmDMZyVK AVVFQtn5…tj9sW3Zp

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.

2.182 × 10¹⁰⁵(106-digit number)

21823635352900099243…37238344348886293929

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

2.182 × 10¹⁰⁵(106-digit number)

21823635352900099243…37238344348886293929

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.182 × 10¹⁰⁵(106-digit number)

21823635352900099243…37238344348886293931

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

4.364 × 10¹⁰⁵(106-digit number)

43647270705800198487…74476688697772587859

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.364 × 10¹⁰⁵(106-digit number)

43647270705800198487…74476688697772587861

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

8.729 × 10¹⁰⁵(106-digit number)

87294541411600396975…48953377395545175719

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.729 × 10¹⁰⁵(106-digit number)

87294541411600396975…48953377395545175721

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

1.745 × 10¹⁰⁶(107-digit number)

17458908282320079395…97906754791090351439

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.745 × 10¹⁰⁶(107-digit number)

17458908282320079395…97906754791090351441

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