Block #159,157

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/10/2013, 10:13:26 PM · Difficulty 9.8657 · 6,649,628 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

3f57dc6ac78fd7ac9b97518ae05e39a817b569d8a1c23d956b9121edd366f4a9

View on Chainz ↗

Height

#159,157

Difficulty

9.865687

Transactions

Size

1.35 KB

Version

Bits

09dd9db1

Nonce

236,344

Timestamp

9/10/2013, 10:13:26 PM

Confirmations

6,649,628

Mined by

⛏️ P2SHANkSBqbEXP5SFNG3bGPJ6iV1UsLajBMwP7

Merkle Root

8bbc8d418c958ffd24d6dcdfce130007431e93ed84f2e548f75803cfe227055e

Transactions (2)

Coinbase610fe29773169d…3a8b123208ebe9

1 in → 1 out10.2800 XPM109 B

ANkSBqbE…LajBMwP7

f9c2d79268632e…3d38190464ecc8

10 in → 1 out102.6100 XPM1.15 KB

AYeYSNLo…NiL9hDLh

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.

1.519 × 10⁹³(94-digit number)

15192342402040857068…27834394024980753279

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

1.519 × 10⁹³(94-digit number)

15192342402040857068…27834394024980753279

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.519 × 10⁹³(94-digit number)

15192342402040857068…27834394024980753281

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

3.038 × 10⁹³(94-digit number)

30384684804081714137…55668788049961506559

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.038 × 10⁹³(94-digit number)

30384684804081714137…55668788049961506561

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

6.076 × 10⁹³(94-digit number)

60769369608163428275…11337576099923013119

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.076 × 10⁹³(94-digit number)

60769369608163428275…11337576099923013121

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

1.215 × 10⁹⁴(95-digit number)

12153873921632685655…22675152199846026239

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.215 × 10⁹⁴(95-digit number)

12153873921632685655…22675152199846026241

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

2.430 × 10⁹⁴(95-digit number)

24307747843265371310…45350304399692052479

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 #159,157

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