Block #269,132

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/22/2013, 7:26:23 PM · Difficulty 9.9548 · 6,541,439 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

5821fc37f6d066a3f8e27cbc7d0852b2331e2c43de276ab60c8e9862529c3183

View on Chainz ↗

Height

#269,132

Difficulty

9.954847

Transactions

Size

2.24 KB

Version

Bits

09f470d7

Nonce

163,471

Timestamp

11/22/2013, 7:26:23 PM

Confirmations

6,541,439

Mined by

⛏️ LaRANHbaxZpBSqLnDhrQC2JecyGXbXjnHCJJs

Merkle Root

bbbcb2cfdaa8e794e4c0615337d3106c300440ee312d9a07fc0b5c71956bc687

Transactions (1)

Coinbasebbbcb2cfdaa8e7…0b5c71956bc687

1 in → 63 out10.0500 XPM2.15 KB

ANHbaxZp…XjnHCJJs AVzhvfTX…ZzNJYq61 AHz4XNRs…v3tseoHZ AVcaNYiv…svWC5rek ANo4YY1x…vnsPRDSU

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.

2.040 × 10⁹¹(92-digit number)

20403113973840718649…84159925082924881159

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

2.040 × 10⁹¹(92-digit number)

20403113973840718649…84159925082924881159

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.040 × 10⁹¹(92-digit number)

20403113973840718649…84159925082924881161

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

4.080 × 10⁹¹(92-digit number)

40806227947681437299…68319850165849762319

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.080 × 10⁹¹(92-digit number)

40806227947681437299…68319850165849762321

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

8.161 × 10⁹¹(92-digit number)

81612455895362874599…36639700331699524639

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.161 × 10⁹¹(92-digit number)

81612455895362874599…36639700331699524641

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

1.632 × 10⁹²(93-digit number)

16322491179072574919…73279400663399049279

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.632 × 10⁹²(93-digit number)

16322491179072574919…73279400663399049281

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

3.264 × 10⁹²(93-digit number)

32644982358145149839…46558801326798098559

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 #269,132

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