Block #403,725

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/14/2014, 9:01:04 AM · Difficulty 10.4318 · 6,400,043 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

975b69abc87c72b1a995094e9689c371a8633f213b85b079e3ed588d06fe02af

View on Chainz ↗

Height

#403,725

Difficulty

10.431787

Transactions

Size

901 B

Version

Bits

0a6e8994

Nonce

121,354

Timestamp

2/14/2014, 9:01:04 AM

Confirmations

6,400,043

Mined by

⛏️ aR;AQvZBsUoAqCaKecR1VEueWTuBchppgRZTJ

Merkle Root

2095caeeaa4ccc2dc75f7b360b405791d032c83b5bb0f3f0a07c836c8b1ab357

Transactions (1)

Coinbase2095caeeaa4ccc…7c836c8b1ab357

1 in → 22 out9.1800 XPM811 B

AQvZBsUo…hppgRZTJ Aac9Hjdg…P4ktypc3 AYRvXuTr…CkbwFeLY AGKhfQo2…h7fBXLX6 AJzWx2VD…j4ZSMit5

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.

3.135 × 10⁹³(94-digit number)

31350312647054987758…30172723691322635519

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

3.135 × 10⁹³(94-digit number)

31350312647054987758…30172723691322635519

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.135 × 10⁹³(94-digit number)

31350312647054987758…30172723691322635521

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

6.270 × 10⁹³(94-digit number)

62700625294109975516…60345447382645271039

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.270 × 10⁹³(94-digit number)

62700625294109975516…60345447382645271041

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

1.254 × 10⁹⁴(95-digit number)

12540125058821995103…20690894765290542079

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.254 × 10⁹⁴(95-digit number)

12540125058821995103…20690894765290542081

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

2.508 × 10⁹⁴(95-digit number)

25080250117643990206…41381789530581084159

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.508 × 10⁹⁴(95-digit number)

25080250117643990206…41381789530581084161

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

5.016 × 10⁹⁴(95-digit number)

50160500235287980413…82763579061162168319

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

5.016 × 10⁹⁴(95-digit number)

50160500235287980413…82763579061162168321

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