Block #267,116

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/21/2013, 12:04:48 AM · Difficulty 9.9597 · 6,540,024 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

35ecbd172079c104fac7418173abb4bdde8720b31e4cf990722006781b2daa62

View on Chainz ↗

Height

#267,116

Difficulty

9.959711

Transactions

Size

2.41 KB

Version

Bits

09f5afa1

Nonce

1,644

Timestamp

11/21/2013, 12:04:48 AM

Confirmations

6,540,024

Mined by

⛏️ P2SHAGssgv8CenT6oWQcceq6rFxSyqzT3wunSb

Merkle Root

25880a0d0997dc0a853c42983dbbd1391c462d8401a51609a1cf6cd772f7fe42

Transactions (2)

Coinbase20fcc6fbf3c450…5b44ee1cb97b17

1 in → 1 out10.1000 XPM109 B

AGssgv8C…zT3wunSb

abaaf5cc8dfc5f…c7eefef9c6bade

18 in → 2 out140.9101 XPM2.21 KB

ARxXxLo4…PYMjUNPs ANdyZLDe…gBkULtj1

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.

9.781 × 10⁹³(94-digit number)

97815074810745402171…24434994741005998079

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

9.781 × 10⁹³(94-digit number)

97815074810745402171…24434994741005998079

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

9.781 × 10⁹³(94-digit number)

97815074810745402171…24434994741005998081

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

1.956 × 10⁹⁴(95-digit number)

19563014962149080434…48869989482011996159

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.956 × 10⁹⁴(95-digit number)

19563014962149080434…48869989482011996161

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

3.912 × 10⁹⁴(95-digit number)

39126029924298160868…97739978964023992319

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.912 × 10⁹⁴(95-digit number)

39126029924298160868…97739978964023992321

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

7.825 × 10⁹⁴(95-digit number)

78252059848596321737…95479957928047984639

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.825 × 10⁹⁴(95-digit number)

78252059848596321737…95479957928047984641

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

1.565 × 10⁹⁵(96-digit number)

15650411969719264347…90959915856095969279

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 #267,116

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