Block #266,699

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/20/2013, 3:52:29 PM · Difficulty 9.9603 · 6,525,119 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

c3cdcde0b07e7af8fef19178b1a7cc3ea6fe6fc6ebf381f96e46d4a655b8527d

View on Chainz ↗

Height

#266,699

Difficulty

9.960296

Transactions

Size

724 B

Version

Bits

09f5d5f2

Nonce

23,084

Timestamp

11/20/2013, 3:52:29 PM

Confirmations

6,525,119

Merkle Root

2630ff8fdb18404b9480e96571669cfa2071b7c2812f5b0f64237b4d1e087ea0

Transactions (2)

Coinbase2c466bf1e37611…dc41ae6db40db5

1 in → 1 out10.0700 XPM110 B

AeKNrjNj…1NEq95Ta

24a26a86e70c6f…59111b04f447df

3 in → 2 out5.5744 XPM522 B

AUyNFf3W…6uzgtGQ9 AQPqCzkF…X4v6zt9y

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.

8.663 × 10⁹⁸(99-digit number)

86630197660791393034…09439784485024255999

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

8.663 × 10⁹⁸(99-digit number)

86630197660791393034…09439784485024255999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.663 × 10⁹⁸(99-digit number)

86630197660791393034…09439784485024256001

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

1.732 × 10⁹⁹(100-digit number)

17326039532158278606…18879568970048511999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.732 × 10⁹⁹(100-digit number)

17326039532158278606…18879568970048512001

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

3.465 × 10⁹⁹(100-digit number)

34652079064316557213…37759137940097023999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.465 × 10⁹⁹(100-digit number)

34652079064316557213…37759137940097024001

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

6.930 × 10⁹⁹(100-digit number)

69304158128633114427…75518275880194047999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.930 × 10⁹⁹(100-digit number)

69304158128633114427…75518275880194048001

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

1.386 × 10¹⁰⁰(101-digit number)

13860831625726622885…51036551760388095999

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