Block #81,768

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 7/24/2013, 11:51:44 PM · Difficulty 9.2701 · 6,709,226 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

42103c1beb6c7883c86f64ad63c8f5f2ae0732312effdd5d814a303cc176c8f9

View on Chainz ↗

Height

#81,768

Difficulty

9.270097

Transactions

Size

947 B

Version

Bits

09452511

Nonce

144

Timestamp

7/24/2013, 11:51:44 PM

Confirmations

6,709,226

Merkle Root

2dda4723ff1c0cafb41310649e7c5200f963ec0d1b177cf8c717ab15b8cde1e1

Transactions (3)

Coinbase2b37d2393e7a88…8b32588fcebc66

1 in → 1 out11.6400 XPM110 B

AXE3Hefj…1xkLbPsS

ea1d740a14f6fb…3f5616df9ecfb1

1 in → 2 out58.2500 XPM225 B

AerNGAw7…UUuKfQwp ATBexGfu…fnF1iiX6

3ae682d1cf0edc…c0b9091aab6e1f

3 in → 2 out657.2700 XPM521 B

APpKmwPq…76A369Z8 AWAubrxZ…5s2sYWGY

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.

3.418 × 10⁹⁶(97-digit number)

34184675419572625119…51871746354991920639

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

3.418 × 10⁹⁶(97-digit number)

34184675419572625119…51871746354991920639

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.418 × 10⁹⁶(97-digit number)

34184675419572625119…51871746354991920641

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

6.836 × 10⁹⁶(97-digit number)

68369350839145250238…03743492709983841279

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.836 × 10⁹⁶(97-digit number)

68369350839145250238…03743492709983841281

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

1.367 × 10⁹⁷(98-digit number)

13673870167829050047…07486985419967682559

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.367 × 10⁹⁷(98-digit number)

13673870167829050047…07486985419967682561

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

2.734 × 10⁹⁷(98-digit number)

27347740335658100095…14973970839935365119

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.734 × 10⁹⁷(98-digit number)

27347740335658100095…14973970839935365121

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

5.469 × 10⁹⁷(98-digit number)

54695480671316200190…29947941679870730239

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 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 9

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