Block #270,451

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/23/2013, 11:10:35 PM · Difficulty 9.9517 · 6,537,438 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

d99d73895832359263a347f94a90de682e13c23f7faafeec5a88c84217f68874

View on Chainz ↗

Height

#270,451

Difficulty

9.951692

Transactions

Size

2.01 KB

Version

Bits

09f3a211

Nonce

43,401

Timestamp

11/23/2013, 11:10:35 PM

Confirmations

6,537,438

Mined by

⛏️ sAbNpwuMpfrkTAQgJoCKTbuG3RcL1hQdb3v

Merkle Root

ec344af84a1c3badc31aaefe08d2b5c1a29a3d9483498695ea7f8aa0d4ae8579

Transactions (1)

Coinbaseec344af84a1c3b…7f8aa0d4ae8579

1 in → 56 out10.0600 XPM1.92 KB

AbNpwuMp…L1hQdb3v AdC8NfKe…9csBTCUH ANHbaxZp…XjnHCJJs AVzhvfTX…ZzNJYq61 AWZd1qVJ…9pj9yfPa

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.

8.896 × 10⁹³(94-digit number)

88961243635381813940…18695710480864182239

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

8.896 × 10⁹³(94-digit number)

88961243635381813940…18695710480864182239

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.896 × 10⁹³(94-digit number)

88961243635381813940…18695710480864182241

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

1.779 × 10⁹⁴(95-digit number)

17792248727076362788…37391420961728364479

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.779 × 10⁹⁴(95-digit number)

17792248727076362788…37391420961728364481

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

3.558 × 10⁹⁴(95-digit number)

35584497454152725576…74782841923456728959

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.558 × 10⁹⁴(95-digit number)

35584497454152725576…74782841923456728961

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

7.116 × 10⁹⁴(95-digit number)

71168994908305451152…49565683846913457919

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.116 × 10⁹⁴(95-digit number)

71168994908305451152…49565683846913457921

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

1.423 × 10⁹⁵(96-digit number)

14233798981661090230…99131367693826915839

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 #270,451

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