Block #51,318

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/16/2013, 5:15:59 AM · Difficulty 8.8967 · 6,738,521 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

4dc5b9c21a243700c9d0e41a6504b45d8dff341935c836d21f3e777ce86e6e35

View on Chainz ↗

Height

#51,318

Difficulty

8.896651

Transactions

Size

13.49 KB

Version

Bits

08e58ae7

Nonce

127

Timestamp

7/16/2013, 5:15:59 AM

Confirmations

6,738,521

Merkle Root

914f6f3c78c7d3d595a3c4b7bd301563ff08c7dd4366600e3149a9c755be93a3

Transactions (4)

Coinbased1555117d4332d…d721a026e0a89d

1 in → 1 out12.7700 XPM110 B

AZzPdrag…DDhMh5eY

fa180cc120d326…e22e0b280d61da

31 in → 2 out400.0100 XPM3.57 KB

ANBGBvzR…CgPGtqfX AeYD7dmk…PUkSHQXJ

743b0a62b49922…1e39af2ea311f7

55 in → 1 out700.0000 XPM6.20 KB

AduwKP5J…wp1LyLxL

4f4686d9bc899f…4592e4700ac4e1

31 in → 2 out400.5300 XPM3.53 KB

Abh159rJ…UE1RykBw AUudveRy…GwH1HbdX

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.

6.220 × 10⁹⁴(95-digit number)

62205244975935496137…65903448591030086879

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

6.220 × 10⁹⁴(95-digit number)

62205244975935496137…65903448591030086879

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.220 × 10⁹⁴(95-digit number)

62205244975935496137…65903448591030086881

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

1.244 × 10⁹⁵(96-digit number)

12441048995187099227…31806897182060173759

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.244 × 10⁹⁵(96-digit number)

12441048995187099227…31806897182060173761

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

2.488 × 10⁹⁵(96-digit number)

24882097990374198455…63613794364120347519

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.488 × 10⁹⁵(96-digit number)

24882097990374198455…63613794364120347521

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

4.976 × 10⁹⁵(96-digit number)

49764195980748396910…27227588728240695039

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.976 × 10⁹⁵(96-digit number)

49764195980748396910…27227588728240695041

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