Block #186,171

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/29/2013, 6:22:02 PM · Difficulty 9.8645 · 6,604,890 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

431d619d57458cbb84628807b0a569c9f452560409a591b2505ff74df6e626b9

View on Chainz ↗

Height

#186,171

Difficulty

9.864547

Transactions

Size

207 B

Version

Bits

09dd52f8

Nonce

117,441,649

Timestamp

9/29/2013, 6:22:02 PM

Confirmations

6,604,890

Merkle Root

31703735c83784d18b6fe64b4ef05bc6f32d0dcee4f2477e03717152e320a71e

Transactions (1)

Coinbase31703735c83784…717152e320a71e

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.

2.591 × 10⁹⁶(97-digit number)

25915664090800879390…55966344748139093119

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

2.591 × 10⁹⁶(97-digit number)

25915664090800879390…55966344748139093119

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.591 × 10⁹⁶(97-digit number)

25915664090800879390…55966344748139093121

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

5.183 × 10⁹⁶(97-digit number)

51831328181601758781…11932689496278186239

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.183 × 10⁹⁶(97-digit number)

51831328181601758781…11932689496278186241

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

1.036 × 10⁹⁷(98-digit number)

10366265636320351756…23865378992556372479

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.036 × 10⁹⁷(98-digit number)

10366265636320351756…23865378992556372481

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

2.073 × 10⁹⁷(98-digit number)

20732531272640703512…47730757985112744959

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.073 × 10⁹⁷(98-digit number)

20732531272640703512…47730757985112744961

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

4.146 × 10⁹⁷(98-digit number)

41465062545281407025…95461515970225489919

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