Block #157,841

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/9/2013, 10:01:29 PM · Difficulty 9.8692 · 6,650,297 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e3159b734019f34cae2e7374f4c70c4b458d1f23548a9823850f37221a143181

View on Chainz ↗

Height

#157,841

Difficulty

9.869244

Transactions

Size

2.58 KB

Version

Bits

09de86cb

Nonce

195,477

Timestamp

9/9/2013, 10:01:29 PM

Confirmations

6,650,297

Mined by

⛏️ P2SHAShFVGaFRKJm4pPMvikZMYpPTLwv2Vsbj7

Merkle Root

2e7d89a3ce47806a3e3297d8476a08eb4cd86d578c1a10791e280b3bc8631961

Transactions (3)

Coinbased348cbc01c6ca7…2807e45ff6656b

1 in → 1 out10.2800 XPM109 B

AShFVGaF…wv2Vsbj7

cfe01dc0173b5f…b0f7c3fb7a87fd

6 in → 2 out299.0761 XPM969 B

AWErAaYS…BW1HFTXN ATr9fboA…4LbjKoQQ

3f7666ed00367d…39245680f00636

12 in → 2 out119.1100 XPM1.44 KB

AGZvBxxa…shhkwtTn ARtAkvkY…FAmm8vmH

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.

8.095 × 10⁹⁴(95-digit number)

80956330521049205012…11028432855454069719

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

8.095 × 10⁹⁴(95-digit number)

80956330521049205012…11028432855454069719

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.095 × 10⁹⁴(95-digit number)

80956330521049205012…11028432855454069721

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

1.619 × 10⁹⁵(96-digit number)

16191266104209841002…22056865710908139439

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.619 × 10⁹⁵(96-digit number)

16191266104209841002…22056865710908139441

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

3.238 × 10⁹⁵(96-digit number)

32382532208419682004…44113731421816278879

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.238 × 10⁹⁵(96-digit number)

32382532208419682004…44113731421816278881

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

6.476 × 10⁹⁵(96-digit number)

64765064416839364009…88227462843632557759

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.476 × 10⁹⁵(96-digit number)

64765064416839364009…88227462843632557761

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

1.295 × 10⁹⁶(97-digit number)

12953012883367872801…76454925687265115519

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 #157,841

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