Block #345,349

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 1/5/2014, 9:12:04 PM · Difficulty 10.2090 · 6,469,546 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

836d523cc2a12fcefdefb800c6a557a8ddc299a0d304621b5721b375cc8e459b

View on Chainz ↗

Height

#345,349

Difficulty

10.209044

Transactions

Size

1.00 KB

Version

Bits

0a3583e2

Nonce

51,004

Timestamp

1/5/2014, 9:12:04 PM

Confirmations

6,469,546

Mined by

⛏️ P2SHAcp7QB7BtGjzejuqs8DZZmn2agw6E8CcS3

Merkle Root

e46405c5d38a1b5a39185197169cc7421e70ae7e4060be05f41dfa9334390b65

Transactions (4)

Coinbasef1f80ed9cd8c1c…dd31430ed74d15

1 in → 1 out9.6100 XPM109 B

Acp7QB7B…w6E8CcS3

137145bbc64270…d1c55cc64979c2

1 in → 2 out3.5776 XPM227 B

ARd47WCo…vXzjgLiX ARGiidg4…RtrYmMww

ea9c8c3af07176…2f503cc12cf5d1

1 in → 2 out6.6429 XPM226 B

AGbLHkXe…Qbq4y5KH ASvgx8Da…89Vb3MWP

3f16917376787e…ad65b6cb6d9dd0

2 in → 2 out5.1097 XPM374 B

ALN4icNX…REmZaGtf ALmRsx4a…ZUp7Wfnr

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.

2.033 × 10⁹⁹(100-digit number)

20333567497421836496…81908684099756317439

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

2.033 × 10⁹⁹(100-digit number)

20333567497421836496…81908684099756317439

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.033 × 10⁹⁹(100-digit number)

20333567497421836496…81908684099756317441

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

4.066 × 10⁹⁹(100-digit number)

40667134994843672993…63817368199512634879

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.066 × 10⁹⁹(100-digit number)

40667134994843672993…63817368199512634881

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

8.133 × 10⁹⁹(100-digit number)

81334269989687345986…27634736399025269759

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.133 × 10⁹⁹(100-digit number)

81334269989687345986…27634736399025269761

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

1.626 × 10¹⁰⁰(101-digit number)

16266853997937469197…55269472798050539519

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.626 × 10¹⁰⁰(101-digit number)

16266853997937469197…55269472798050539521

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

3.253 × 10¹⁰⁰(101-digit number)

32533707995874938394…10538945596101079039

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.253 × 10¹⁰⁰(101-digit number)

32533707995874938394…10538945596101079041

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 #345,349

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