Block #265,998

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 11/20/2013, 2:16:40 AM · Difficulty 9.9612 · 6,544,272 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

3ab65696660517fa8bd874bfb60552c223db3d4c3df41997c88bbd1747d5e7f7

View on Chainz ↗

Height

#265,998

Difficulty

9.961168

Transactions

Size

1.84 KB

Version

Bits

09f60f18

Nonce

28,343

Timestamp

11/20/2013, 2:16:40 AM

Confirmations

6,544,272

Mined by

⛏️ aRASk3VNLyMBGpzTmUCmX3xNhd2H13sqXPzG

Merkle Root

b42a0635cec1dc9eae41f9d2403bfea6f0300f4ebf52941237979f314f3f9177

Transactions (1)

Coinbaseb42a0635cec1dc…979f314f3f9177

1 in → 51 out10.0400 XPM1.75 KB

ASk3VNLy…13sqXPzG ALgGDFyg…tHka2xUd APZy8y6E…QZaGQjfp ANHbaxZp…XjnHCJJs AVzhvfTX…ZzNJYq61

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.

1.428 × 10⁹⁷(98-digit number)

14286264018590647000…33676722039845159999

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

1.428 × 10⁹⁷(98-digit number)

14286264018590647000…33676722039845159999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.428 × 10⁹⁷(98-digit number)

14286264018590647000…33676722039845160001

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

2.857 × 10⁹⁷(98-digit number)

28572528037181294001…67353444079690319999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.857 × 10⁹⁷(98-digit number)

28572528037181294001…67353444079690320001

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

5.714 × 10⁹⁷(98-digit number)

57145056074362588003…34706888159380639999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.714 × 10⁹⁷(98-digit number)

57145056074362588003…34706888159380640001

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

1.142 × 10⁹⁸(99-digit number)

11429011214872517600…69413776318761279999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.142 × 10⁹⁸(99-digit number)

11429011214872517600…69413776318761280001

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

2.285 × 10⁹⁸(99-digit number)

22858022429745035201…38827552637522559999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.285 × 10⁹⁸(99-digit number)

22858022429745035201…38827552637522560001

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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 #265,998

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