Block #142,166

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 8/30/2013, 7:18:10 PM · Difficulty 9.8361 · 6,672,703 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

fbe173df3d9f5ecaff028ef9c1a0934c5c932c48eb17ac753f5146cff522529f

View on Chainz ↗

Height

#142,166

Difficulty

9.836144

Transactions

Size

574 B

Version

Bits

09d60d85

Nonce

12,303

Timestamp

8/30/2013, 7:18:10 PM

Confirmations

6,672,703

Mined by

⛏️ P2SHAZaviS5niSRu6FiSfjS3FTpg9coHuQufdR

Merkle Root

a0f810d7a21419bb48995421d98829d859144230e457df27f82993769506c557

Transactions (2)

Coinbasede90aa0aaa2554…a11b48b3bf8fc7

1 in → 1 out10.3300 XPM109 B

AZaviS5n…oHuQufdR

07603fefd028ef…d915a4434f597d

2 in → 2 out1.7300 XPM375 B

AeMmfENn…jNrRQskV Abtjhqm9…Ggn5PnWJ

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.

6.876 × 10⁹³(94-digit number)

68765815923816891832…63128894298953432559

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

6.876 × 10⁹³(94-digit number)

68765815923816891832…63128894298953432559

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.876 × 10⁹³(94-digit number)

68765815923816891832…63128894298953432561

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

1.375 × 10⁹⁴(95-digit number)

13753163184763378366…26257788597906865119

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.375 × 10⁹⁴(95-digit number)

13753163184763378366…26257788597906865121

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

2.750 × 10⁹⁴(95-digit number)

27506326369526756732…52515577195813730239

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.750 × 10⁹⁴(95-digit number)

27506326369526756732…52515577195813730241

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

5.501 × 10⁹⁴(95-digit number)

55012652739053513465…05031154391627460479

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.501 × 10⁹⁴(95-digit number)

55012652739053513465…05031154391627460481

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

1.100 × 10⁹⁵(96-digit number)

11002530547810702693…10062308783254920959

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

Block #142,166

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