Block #413,520

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/21/2014, 7:21:12 AM · Difficulty 10.4152 · 6,390,038 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

9bea762fe1d007c409040584e9cdb92ac32b3cc48ab2fe337e6acdc2514022e2

View on Chainz ↗

Height

#413,520

Difficulty

10.415222

Transactions

Size

902 B

Version

Bits

0a6a4bf5

Nonce

6,577

Timestamp

2/21/2014, 7:21:12 AM

Confirmations

6,390,038

Mined by

⛏️ POaS\AQvZBsUoAqCaKecR1VEueWTuBchppgRZTJ

Merkle Root

b8915480a9133dc8734b7632f00df0807f5f9c4a360c210759e0afd842f5094b

Transactions (1)

Coinbaseb8915480a9133d…e0afd842f5094b

1 in → 22 out9.2000 XPM811 B

AQvZBsUo…hppgRZTJ AQwRN8bE…V8PsTcY9 Aac9Hjdg…P4ktypc3 AZV4TwxQ…8JnQqSoH AGKhfQo2…h7fBXLX6

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.

2.333 × 10⁹⁶(97-digit number)

23338747816297648348…46056845404737438239

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

2.333 × 10⁹⁶(97-digit number)

23338747816297648348…46056845404737438239

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.333 × 10⁹⁶(97-digit number)

23338747816297648348…46056845404737438241

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

4.667 × 10⁹⁶(97-digit number)

46677495632595296696…92113690809474876479

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.667 × 10⁹⁶(97-digit number)

46677495632595296696…92113690809474876481

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

9.335 × 10⁹⁶(97-digit number)

93354991265190593392…84227381618949752959

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.335 × 10⁹⁶(97-digit number)

93354991265190593392…84227381618949752961

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

1.867 × 10⁹⁷(98-digit number)

18670998253038118678…68454763237899505919

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.867 × 10⁹⁷(98-digit number)

18670998253038118678…68454763237899505921

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

3.734 × 10⁹⁷(98-digit number)

37341996506076237356…36909526475799011839

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.734 × 10⁹⁷(98-digit number)

37341996506076237356…36909526475799011841

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