Block #186,146

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/29/2013, 5:57:44 PM · Difficulty 9.8645 · 6,605,406 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

bda5ee6b1c4613ffd0958919bd49ec5f87203f0ea44d18bea5a81a3e8036a4f2

View on Chainz ↗

Height

#186,146

Difficulty

9.864550

Transactions

Size

204 B

Version

Bits

09dd5321

Nonce

33,554,496

Timestamp

9/29/2013, 5:57:44 PM

Confirmations

6,605,406

Merkle Root

3c41f962ddf860b666ea1bb8ded5f883f6871ef09959dc11c4aa8807b62a7ce0

Transactions (1)

Coinbase3c41f962ddf860…aa8807b62a7ce0

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.

4.109 × 10⁸⁹(90-digit number)

41091199993051601606…59092418113345220549

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

4.109 × 10⁸⁹(90-digit number)

41091199993051601606…59092418113345220549

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.109 × 10⁸⁹(90-digit number)

41091199993051601606…59092418113345220551

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

8.218 × 10⁸⁹(90-digit number)

82182399986103203212…18184836226690441099

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.218 × 10⁸⁹(90-digit number)

82182399986103203212…18184836226690441101

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

1.643 × 10⁹⁰(91-digit number)

16436479997220640642…36369672453380882199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.643 × 10⁹⁰(91-digit number)

16436479997220640642…36369672453380882201

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

3.287 × 10⁹⁰(91-digit number)

32872959994441281284…72739344906761764399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.287 × 10⁹⁰(91-digit number)

32872959994441281284…72739344906761764401

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

6.574 × 10⁹⁰(91-digit number)

65745919988882562569…45478689813523528799

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