Block #166,586

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/16/2013, 12:30:12 AM · Difficulty 9.8684 · 6,649,720 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

5ace2a1421c001aa942e8b8467b201d63f3f9a01166c2ebb52450aea23f28a85

View on Chainz ↗

Height

#166,586

Difficulty

9.868433

Transactions

Size

1.36 KB

Version

Bits

09de5199

Nonce

124,653

Timestamp

9/16/2013, 12:30:12 AM

Confirmations

6,649,720

Mined by

⛏️ P2SHAMa6LrBTTD2fuQYWRy4RSzDmpYpGxPwh7T

Merkle Root

ad7077d9770d8d6f7933447b8906780490b2b7e2a5f70d78b44c01e15c76e9f6

Transactions (4)

Coinbase7b35f90f75ea9a…c75ea82ea421e5

1 in → 1 out10.2800 XPM109 B

AMa6LrBT…pGxPwh7T

dd9f1a12113d3f…2fcb66de345078

1 in → 2 out4.9800 XPM226 B

AbMYhG3U…RrRrsaq8 AHupvmLz…FZHW4z6p

180ae991f21a34…d90136cbf5959a

1 in → 2 out3.9700 XPM227 B

AcqUrck5…Lbczpyi6 AWcZjGw7…FWC6X8Wu

6fd5a11ed8377b…59fd1308b1f729

1 in → 17 out33.5500 XPM735 B

AQSQ3ADY…fvDboxSd AbU9ZqFp…T7DELSpa ATv4Kcjt…k5rHgkt6 AMdLuRsK…1VBNjXzx AWUfiqwg…j38kk5cK

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.

1.083 × 10⁹⁶(97-digit number)

10830052439973326995…88153629543619839499

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

1.083 × 10⁹⁶(97-digit number)

10830052439973326995…88153629543619839499

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.083 × 10⁹⁶(97-digit number)

10830052439973326995…88153629543619839501

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

2.166 × 10⁹⁶(97-digit number)

21660104879946653990…76307259087239678999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.166 × 10⁹⁶(97-digit number)

21660104879946653990…76307259087239679001

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

4.332 × 10⁹⁶(97-digit number)

43320209759893307981…52614518174479357999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.332 × 10⁹⁶(97-digit number)

43320209759893307981…52614518174479358001

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

8.664 × 10⁹⁶(97-digit number)

86640419519786615963…05229036348958715999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

8.664 × 10⁹⁶(97-digit number)

86640419519786615963…05229036348958716001

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

1.732 × 10⁹⁷(98-digit number)

17328083903957323192…10458072697917431999

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

Block #166,586

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