Block #409,821

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/18/2014, 3:23:05 PM · Difficulty 10.4292 · 6,394,373 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

b62295c1d12125e5b2042c04717056c8af562463b5540141e07b58293e17f8c4

View on Chainz ↗

Height

#409,821

Difficulty

10.429245

Transactions

Size

391 B

Version

Bits

0a6de306

Nonce

293,700

Timestamp

2/18/2014, 3:23:05 PM

Confirmations

6,394,373

Mined by

⛏️ P2SHAKCRCCETW8FUumyspNEmHXiVqDqxdRnend

Merkle Root

3e507d71844fffeab35206d2cfc2c68036d93bb815fdc7d9d2dd048ec94a1ce2

Transactions (2)

Coinbasea8c76ad4a6d67a…320bc1d9323fce

1 in → 1 out9.1900 XPM109 B

AKCRCCET…qxdRnend

1dc2d81d372b8e…e5da897797b52a

1 in → 1 out1.9955 XPM191 B

AR6hsGeo…KaU2e4uJ

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.

5.614 × 10⁹⁷(98-digit number)

56140563968755260963…18185934102305384379

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

5.614 × 10⁹⁷(98-digit number)

56140563968755260963…18185934102305384379

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.614 × 10⁹⁷(98-digit number)

56140563968755260963…18185934102305384381

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

1.122 × 10⁹⁸(99-digit number)

11228112793751052192…36371868204610768759

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.122 × 10⁹⁸(99-digit number)

11228112793751052192…36371868204610768761

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

2.245 × 10⁹⁸(99-digit number)

22456225587502104385…72743736409221537519

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.245 × 10⁹⁸(99-digit number)

22456225587502104385…72743736409221537521

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

4.491 × 10⁹⁸(99-digit number)

44912451175004208770…45487472818443075039

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.491 × 10⁹⁸(99-digit number)

44912451175004208770…45487472818443075041

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

8.982 × 10⁹⁸(99-digit number)

89824902350008417540…90974945636886150079

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

8.982 × 10⁹⁸(99-digit number)

89824902350008417540…90974945636886150081

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