Block #414,342

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/21/2014, 10:28:02 PM · Difficulty 10.4054 · 6,382,540 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

9988d68771a3e2a3409787c4dd12f1305242c03a8acbd611b8ef21a8366496fe

View on Chainz ↗

Height

#414,342

Difficulty

10.405403

Transactions

Size

968 B

Version

Bits

0a67c879

Nonce

9,875

Timestamp

2/21/2014, 10:28:02 PM

Confirmations

6,382,540

Mined by

⛏️ RaSAQvZBsUoAqCaKecR1VEueWTuBchppgRZTJ

Merkle Root

2159ebf34b24d2d57a8808cf17842011d7e84af80780c67636541d5ad1fa4bce

Transactions (1)

Coinbase2159ebf34b24d2…541d5ad1fa4bce

1 in → 24 out9.2200 XPM879 B

AQvZBsUo…hppgRZTJ AJLB8H7s…MRD4s5wA AGKhfQo2…h7fBXLX6 ALqxX1WA…hsKjbnXn ATKK7iFz…wZm46KN1

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.

1.219 × 10⁹³(94-digit number)

12193188862537052821…72109989121497013369

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

1.219 × 10⁹³(94-digit number)

12193188862537052821…72109989121497013369

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.219 × 10⁹³(94-digit number)

12193188862537052821…72109989121497013371

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

2.438 × 10⁹³(94-digit number)

24386377725074105643…44219978242994026739

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.438 × 10⁹³(94-digit number)

24386377725074105643…44219978242994026741

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

4.877 × 10⁹³(94-digit number)

48772755450148211287…88439956485988053479

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.877 × 10⁹³(94-digit number)

48772755450148211287…88439956485988053481

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

9.754 × 10⁹³(94-digit number)

97545510900296422574…76879912971976106959

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

9.754 × 10⁹³(94-digit number)

97545510900296422574…76879912971976106961

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

1.950 × 10⁹⁴(95-digit number)

19509102180059284514…53759825943952213919

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.950 × 10⁹⁴(95-digit number)

19509102180059284514…53759825943952213921

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