Block #186,111

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/29/2013, 5:32:00 PM · Difficulty 9.8639 · 6,610,528 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

24f1e42eeedc8e50741ac71b9d4be08b6f9ca522dc3f26a052961d570bcd9742

View on Chainz ↗

Height

#186,111

Difficulty

9.863947

Transactions

Size

209 B

Version

Bits

09dd2ba6

Nonce

162,908

Timestamp

9/29/2013, 5:32:00 PM

Confirmations

6,610,528

Mined by

⛏️ P2SHAcLBm5Z9BD7o7btAchFh9KZEkSS683d7c3

Merkle Root

39cfaf213baf5c1ed96cc9b1c87c3bafbdce9b0f2cf1acedf5c4b608c440cd43

Transactions (1)

Coinbase39cfaf213baf5c…c4b608c440cd43

1 in → 1 out10.2600 XPM116 B

AcLBm5Z9…S683d7c3

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.

4.094 × 10¹⁰¹(102-digit number)

40942462769646687500…31672460986788464359

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

4.094 × 10¹⁰¹(102-digit number)

40942462769646687500…31672460986788464359

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.094 × 10¹⁰¹(102-digit number)

40942462769646687500…31672460986788464361

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

8.188 × 10¹⁰¹(102-digit number)

81884925539293375001…63344921973576928719

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.188 × 10¹⁰¹(102-digit number)

81884925539293375001…63344921973576928721

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

1.637 × 10¹⁰²(103-digit number)

16376985107858675000…26689843947153857439

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.637 × 10¹⁰²(103-digit number)

16376985107858675000…26689843947153857441

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

3.275 × 10¹⁰²(103-digit number)

32753970215717350000…53379687894307714879

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.275 × 10¹⁰²(103-digit number)

32753970215717350000…53379687894307714881

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

6.550 × 10¹⁰²(103-digit number)

65507940431434700001…06759375788615429759

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