Block #51,769

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/16/2013, 7:21:00 AM · Difficulty 8.9031 · 6,738,176 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

1cd90d80ce0c0d3748942a2e51692ed6eb461a4431859fbeb10aa4afc2401f69

View on Chainz ↗

Height

#51,769

Difficulty

8.903116

Transactions

Size

201 B

Version

Bits

08e732a1

Nonce

912

Timestamp

7/16/2013, 7:21:00 AM

Confirmations

6,738,176

Merkle Root

2bdea0ba9fdabea7e626edfd883704a78da9435e94b6e6c5bb59c4be30cdd603

Transactions (1)

Coinbase2bdea0ba9fdabe…59c4be30cdd603

1 in → 1 out12.6000 XPM110 B

AeTcmmqH…LrFchfe1

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.

2.004 × 10⁹⁶(97-digit number)

20045263450491916893…45070701497531861129

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

2.004 × 10⁹⁶(97-digit number)

20045263450491916893…45070701497531861129

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.004 × 10⁹⁶(97-digit number)

20045263450491916893…45070701497531861131

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

4.009 × 10⁹⁶(97-digit number)

40090526900983833787…90141402995063722259

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.009 × 10⁹⁶(97-digit number)

40090526900983833787…90141402995063722261

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

8.018 × 10⁹⁶(97-digit number)

80181053801967667575…80282805990127444519

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.018 × 10⁹⁶(97-digit number)

80181053801967667575…80282805990127444521

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

1.603 × 10⁹⁷(98-digit number)

16036210760393533515…60565611980254889039

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.603 × 10⁹⁷(98-digit number)

16036210760393533515…60565611980254889041

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