Block #409,721

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/18/2014, 1:52:50 PM · Difficulty 10.4280 · 6,405,227 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f04dfabbb99fdb02f448e189c3457e32e07d34fc04c7e77b9c68ba4e1ce2a7c2

View on Chainz ↗

Height

#409,721

Difficulty

10.428039

Transactions

Size

834 B

Version

Bits

0a6d93fa

Nonce

123,391

Timestamp

2/18/2014, 1:52:50 PM

Confirmations

6,405,227

Mined by

⛏️ y@aSe+aAQvZBsUoAqCaKecR1VEueWTuBchppgRZTJ

Merkle Root

4fa1543466c3c3b78fe920cb5271cb88bd641d024ab1e43a3df3ff67a041dabf

Transactions (1)

Coinbase4fa1543466c3c3…f3ff67a041dabf

1 in → 20 out9.1800 XPM743 B

AQvZBsUo…hppgRZTJ Aac9Hjdg…P4ktypc3 AGKhfQo2…h7fBXLX6 AQwRN8bE…V8PsTcY9 AGLTRtYT…dmN7zcS3

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.

6.848 × 10⁹⁷(98-digit number)

68488434500043833940…05416065536663091199

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

6.848 × 10⁹⁷(98-digit number)

68488434500043833940…05416065536663091199

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.848 × 10⁹⁷(98-digit number)

68488434500043833940…05416065536663091201

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

1.369 × 10⁹⁸(99-digit number)

13697686900008766788…10832131073326182399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.369 × 10⁹⁸(99-digit number)

13697686900008766788…10832131073326182401

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

2.739 × 10⁹⁸(99-digit number)

27395373800017533576…21664262146652364799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.739 × 10⁹⁸(99-digit number)

27395373800017533576…21664262146652364801

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

5.479 × 10⁹⁸(99-digit number)

54790747600035067152…43328524293304729599

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.479 × 10⁹⁸(99-digit number)

54790747600035067152…43328524293304729601

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

1.095 × 10⁹⁹(100-digit number)

10958149520007013430…86657048586609459199

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.095 × 10⁹⁹(100-digit number)

10958149520007013430…86657048586609459201

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

Block #409,721

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