Block #405,851

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/15/2014, 8:23:27 PM · Difficulty 10.4334 · 6,409,257 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

b9c06a2c4734ea09dab0b295103082a04f4bd4f19849214326dceed31885e86f

View on Chainz ↗

Height

#405,851

Difficulty

10.433398

Transactions

Size

1.61 KB

Version

Bits

0a6ef324

Nonce

14,084

Timestamp

2/15/2014, 8:23:27 PM

Confirmations

6,409,257

Mined by

AMW1p9oTkBzM1hi3okaKfJzfWi2TzdaZxH

Merkle Root

467400d56b67407956433a9406bdfb2fde0d6913564255a9b45badc4fc47a2e0

Transactions (4)

Coinbase8589e52a5ebd4f…14b97f8c1ef1ab

1 in → 24 out9.2000 XPM879 B

AMW1p9oT…2TzdaZxH ATp4jJhs…TbSgR8Qb APJPNQaM…8nyTxzkW AcgDwGo8…BBFwbjVo AczyTBWn…bEQQj33u

ea46b9adf05f12…00b89cde582c68

1 in → 2 out4.1384 XPM226 B

AaSEAkF7…2HRinQXG AXHTEx2r…MAW1TDoA

7a54e0a76ca843…e642338a394a3f

1 in → 2 out4.1035 XPM227 B

AZJe4Bxu…Ztr9u4dc AVdZKH55…miZUnrTn

4e0aadadde8cd6…9000351db0a585

1 in → 2 out2.0281 XPM226 B

AMU8nEj9…Eo32pqxr AKsYqscW…2p1GwBJb

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.

8.777 × 10⁹⁹(100-digit number)

87772685092390039294…41822024405705564159

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

8.777 × 10⁹⁹(100-digit number)

87772685092390039294…41822024405705564159

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.777 × 10⁹⁹(100-digit number)

87772685092390039294…41822024405705564161

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

1.755 × 10¹⁰⁰(101-digit number)

17554537018478007858…83644048811411128319

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.755 × 10¹⁰⁰(101-digit number)

17554537018478007858…83644048811411128321

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

3.510 × 10¹⁰⁰(101-digit number)

35109074036956015717…67288097622822256639

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.510 × 10¹⁰⁰(101-digit number)

35109074036956015717…67288097622822256641

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

7.021 × 10¹⁰⁰(101-digit number)

70218148073912031435…34576195245644513279

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.021 × 10¹⁰⁰(101-digit number)

70218148073912031435…34576195245644513281

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

1.404 × 10¹⁰¹(102-digit number)

14043629614782406287…69152390491289026559

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.404 × 10¹⁰¹(102-digit number)

14043629614782406287…69152390491289026561

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 #405,851

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