Block #41,140

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/14/2013, 4:09:12 PM · Difficulty 8.4986 · 6,750,173 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

76540fd60528cd56e624ce4c1eb864394f3f4cad31805b7f4ed91d2d8537df68

View on Chainz ↗

Height

#41,140

Difficulty

8.498570

Transactions

Size

361 B

Version

Bits

087fa248

Nonce

1,001

Timestamp

7/14/2013, 4:09:12 PM

Confirmations

6,750,173

Merkle Root

16d3eeec8f3cb7314e9baa109d1bbb46773aae21265c220296d1cd488f79075b

Transactions (2)

Coinbased62e1ac13c4b26…99cdcb5625f7da

1 in → 1 out13.8400 XPM110 B

AeuKEfPd…zCVKqixu

93ebc7d90545be…d72ed900f665d7

1 in → 1 out15.6200 XPM159 B

AG43TCYA…UW1AUz7e

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.

1.824 × 10⁹⁹(100-digit number)

18245754056082582854…09984904352220575739

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

1.824 × 10⁹⁹(100-digit number)

18245754056082582854…09984904352220575739

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.824 × 10⁹⁹(100-digit number)

18245754056082582854…09984904352220575741

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

3.649 × 10⁹⁹(100-digit number)

36491508112165165709…19969808704441151479

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.649 × 10⁹⁹(100-digit number)

36491508112165165709…19969808704441151481

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

7.298 × 10⁹⁹(100-digit number)

72983016224330331419…39939617408882302959

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

7.298 × 10⁹⁹(100-digit number)

72983016224330331419…39939617408882302961

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

1.459 × 10¹⁰⁰(101-digit number)

14596603244866066283…79879234817764605919

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.459 × 10¹⁰⁰(101-digit number)

14596603244866066283…79879234817764605921

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 8 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

CommonChain length 8

Found in most blocks. The baseline for Primecoin's proof-of-work.

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