Block #165,572

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/15/2013, 9:17:58 AM · Difficulty 9.8657 · 6,630,491 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

578703cdfec6a0865742c6d887b971cae85825e05c4572e4c84ee07fd9b3e2d7

View on Chainz ↗

Height

#165,572

Difficulty

9.865672

Transactions

Size

1.58 KB

Version

Bits

09dd9cab

Nonce

77,958

Timestamp

9/15/2013, 9:17:58 AM

Confirmations

6,630,491

Mined by

⛏️ P2SHAe3zGYLMALYXsJMKVp3DTXRBfHnjrgU7hu

Merkle Root

acc6efb4ad34bdf6f44c587e391100756c52536acecd7a9540e20845764f4f5f

Transactions (4)

Coinbase06364b48084653…11b669e648b7dc

1 in → 1 out10.3000 XPM109 B

Ae3zGYLM…njrgU7hu

2c66496f00991d…9ccd142d9641fb

4 in → 2 out0.6556 XPM670 B

AQE9WSbN…53CijeZj AReea43w…UEqPF1Lp

031faaab5bd90a…6a46d1ab0efd18

2 in → 2 out0.1750 XPM375 B

Ac2LfpDs…kCcmFxDS AMtLvyhH…QzfLCpZS

95e3c6acd7b364…be1cc2cd787c64

2 in → 2 out0.3345 XPM375 B

AYcfAxBq…x6W5agvr AHMo39GL…iVQT9Gn6

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.

7.773 × 10⁹¹(92-digit number)

77736986199987402951…89622243558661398719

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

7.773 × 10⁹¹(92-digit number)

77736986199987402951…89622243558661398719

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.773 × 10⁹¹(92-digit number)

77736986199987402951…89622243558661398721

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

1.554 × 10⁹²(93-digit number)

15547397239997480590…79244487117322797439

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.554 × 10⁹²(93-digit number)

15547397239997480590…79244487117322797441

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

3.109 × 10⁹²(93-digit number)

31094794479994961180…58488974234645594879

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.109 × 10⁹²(93-digit number)

31094794479994961180…58488974234645594881

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

6.218 × 10⁹²(93-digit number)

62189588959989922361…16977948469291189759

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.218 × 10⁹²(93-digit number)

62189588959989922361…16977948469291189761

Verify on FactorDB ↗Wolfram Alpha ↗

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

Level 4 — Twin Prime Pair (2^4 × origin ± 1)

2^4 × origin − 1

1.243 × 10⁹³(94-digit number)

12437917791997984472…33955896938582379519

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 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 9

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