Block #415,717

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/22/2014, 10:27:10 PM · Difficulty 10.3989 · 6,376,101 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

4b32453dbf2e8c5e17272b8fcf8f9ffe48621bee2685cc7895c0a6c19c5624ae

View on Chainz ↗

Height

#415,717

Difficulty

10.398906

Transactions

Size

2.12 KB

Version

Bits

0a661ebc

Nonce

215,470

Timestamp

2/22/2014, 10:27:10 PM

Confirmations

6,376,101

Merkle Root

b9375a849894513d86ec7e7cd8d95841a8c76690d61bd5306fcb55ff3cafc281

Transactions (2)

Coinbasedfa834c1498acd…5964cc16426854

1 in → 22 out9.2300 XPM811 B

AQvZBsUo…hppgRZTJ Aac9Hjdg…P4ktypc3 AYtDvcGs…VuAcA7m7 AGKhfQo2…h7fBXLX6 ALqxX1WA…hsKjbnXn

23dac1ddef1370…30b434332495af

6 in → 17 out55.2200 XPM1.24 KB

AK9KZGFQ…zqoLqy1A AVXqAjBB…2RxF5Hqg AGWSBZZe…bFmzD79i AYKiwoLg…2bSJSJ59 AQYQT37a…woJiw19q

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.

3.061 × 10⁹³(94-digit number)

30615897827242151644…05457336498612515329

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

3.061 × 10⁹³(94-digit number)

30615897827242151644…05457336498612515329

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.061 × 10⁹³(94-digit number)

30615897827242151644…05457336498612515331

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

6.123 × 10⁹³(94-digit number)

61231795654484303288…10914672997225030659

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.123 × 10⁹³(94-digit number)

61231795654484303288…10914672997225030661

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

1.224 × 10⁹⁴(95-digit number)

12246359130896860657…21829345994450061319

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.224 × 10⁹⁴(95-digit number)

12246359130896860657…21829345994450061321

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

2.449 × 10⁹⁴(95-digit number)

24492718261793721315…43658691988900122639

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.449 × 10⁹⁴(95-digit number)

24492718261793721315…43658691988900122641

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

4.898 × 10⁹⁴(95-digit number)

48985436523587442631…87317383977800245279

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.898 × 10⁹⁴(95-digit number)

48985436523587442631…87317383977800245281

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