Block #188,159

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/30/2013, 11:58:29 PM · Difficulty 9.8698 · 6,639,070 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

a51be57caa7b50d1cb6c2158aa55cb01ab9c9a6b37e5a82aff37ebe99451e576

View on Chainz ↗

Height

#188,159

Difficulty

9.869790

Transactions

Size

651 B

Version

Bits

09deaa92

Nonce

226,091

Timestamp

9/30/2013, 11:58:29 PM

Confirmations

6,639,070

Mined by

⛏️ P2SHAQX2SCiqTGkt9bwNKTvA3occxy9uFfUmE1

Merkle Root

ee07a0106f99d59b27ad531b647cb72ca02dfc45ec5164a67aaa85b163404b93

Transactions (3)

Coinbasebb5f3df68ce397…4102449896d62c

1 in → 1 out10.2700 XPM109 B

AQX2SCiq…9uFfUmE1

e5be9a200b743e…42a2e4d66598da

1 in → 2 out10.2600 XPM226 B

AbgoL3dv…RTW7tStT AHyHEeYQ…XuELsExj

2d6e39b7b0610b…b56303b19cb133

1 in → 2 out6.1848 XPM225 B

Ad6DApYk…2A99drc6 AdCbHMuV…xwQmtx9z

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.

1.248 × 10⁹⁷(98-digit number)

12489488856455002628…13728401626307819519

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

1.248 × 10⁹⁷(98-digit number)

12489488856455002628…13728401626307819519

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.248 × 10⁹⁷(98-digit number)

12489488856455002628…13728401626307819521

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

2.497 × 10⁹⁷(98-digit number)

24978977712910005256…27456803252615639039

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.497 × 10⁹⁷(98-digit number)

24978977712910005256…27456803252615639041

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

4.995 × 10⁹⁷(98-digit number)

49957955425820010513…54913606505231278079

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.995 × 10⁹⁷(98-digit number)

49957955425820010513…54913606505231278081

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

9.991 × 10⁹⁷(98-digit number)

99915910851640021026…09827213010462556159

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

9.991 × 10⁹⁷(98-digit number)

99915910851640021026…09827213010462556161

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

1.998 × 10⁹⁸(99-digit number)

19983182170328004205…19654426020925112319

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 #188,159

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