Block #413,931

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/21/2014, 2:10:32 PM · Difficulty 10.4151 · 6,378,766 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

6b9576f0cb50ddaecfb061b31061b83b3ce0b6e16eae79b5417d9f1df37c95a7

View on Chainz ↗

Height

#413,931

Difficulty

10.415094

Transactions

Size

191 B

Version

Bits

0a6a4397

Nonce

102,208

Timestamp

2/21/2014, 2:10:32 PM

Confirmations

6,378,766

Merkle Root

aa9576f64e2e3b50f6ff3d6af803f5d05e17700e824b3f5f7827ecbe606c898f

Transactions (1)

Coinbaseaa9576f64e2e3b…27ecbe606c898f

1 in → 1 out9.2000 XPM100 B

AaXb6Ldz…XefG91jF

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.

5.455 × 10⁹⁶(97-digit number)

54550468309465427828…23065079816189962759

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

5.455 × 10⁹⁶(97-digit number)

54550468309465427828…23065079816189962759

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.455 × 10⁹⁶(97-digit number)

54550468309465427828…23065079816189962761

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

1.091 × 10⁹⁷(98-digit number)

10910093661893085565…46130159632379925519

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.091 × 10⁹⁷(98-digit number)

10910093661893085565…46130159632379925521

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

2.182 × 10⁹⁷(98-digit number)

21820187323786171131…92260319264759851039

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.182 × 10⁹⁷(98-digit number)

21820187323786171131…92260319264759851041

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

4.364 × 10⁹⁷(98-digit number)

43640374647572342263…84520638529519702079

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.364 × 10⁹⁷(98-digit number)

43640374647572342263…84520638529519702081

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

8.728 × 10⁹⁷(98-digit number)

87280749295144684526…69041277059039404159

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

8.728 × 10⁹⁷(98-digit number)

87280749295144684526…69041277059039404161

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