Block #41,154

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/14/2013, 4:10:34 PM · Difficulty 8.4998 · 6,748,571 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

3c2ef2a932da5ecb4848a77bae836c5758991382156353e545e857f6e1072865

View on Chainz ↗

Height

#41,154

Difficulty

8.499827

Transactions

Size

366 B

Version

Bits

087ff4af

Nonce

Timestamp

7/14/2013, 4:10:34 PM

Confirmations

6,748,571

Merkle Root

fef10c21044edd1be07f08300acf4960ddbf3a473578e48c580ca5fff5261964

Transactions (2)

Coinbasecfa12eaedf170b…d7dc54acd61476

1 in → 1 out13.8300 XPM110 B

AVWnWtMv…DtGS9CYs

73970fe961f4f3…4a75953a87e8ef

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.

2.299 × 10¹¹²(113-digit number)

22991960091165699322…06595584405522702499

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

2.299 × 10¹¹²(113-digit number)

22991960091165699322…06595584405522702499

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.299 × 10¹¹²(113-digit number)

22991960091165699322…06595584405522702501

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

4.598 × 10¹¹²(113-digit number)

45983920182331398644…13191168811045404999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.598 × 10¹¹²(113-digit number)

45983920182331398644…13191168811045405001

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

9.196 × 10¹¹²(113-digit number)

91967840364662797289…26382337622090809999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.196 × 10¹¹²(113-digit number)

91967840364662797289…26382337622090810001

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

1.839 × 10¹¹³(114-digit number)

18393568072932559457…52764675244181619999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.839 × 10¹¹³(114-digit number)

18393568072932559457…52764675244181620001

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