Block #212,698

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/16/2013, 11:01:53 AM · Difficulty 9.9194 · 6,585,519 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

44b7b541445ef6cf07bfd4b032f28f01708ef5455bfab5ee941e74a5bb8129c6

View on Chainz ↗

Height

#212,698

Difficulty

9.919354

Transactions

Size

4.80 KB

Version

Bits

09eb5ac4

Nonce

1,164,834,560

Timestamp

10/16/2013, 11:01:53 AM

Confirmations

6,585,519

Mined by

⛏️ >R^rAbdqpFgmN6ooQ1qcRqkeX1J4AKM8UVG4YE

Merkle Root

4c60fac66e07bd93855cc0a358b916f4113884661cc1a3a75f474b022322c590

Transactions (1)

Coinbase4c60fac66e07bd…474b022322c590

1 in → 140 out10.0902 XPM4.71 KB

AbdqpFgm…M8UVG4YE AM4zvBDn…dBXWgwDw AWD7AyMf…5K5HDELu ARVYbwKx…uord52yW ARYXExTc…NwNWrJpG

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.

3.426 × 10⁹⁸(99-digit number)

34264827329559882478…75126927560417612799

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

3.426 × 10⁹⁸(99-digit number)

34264827329559882478…75126927560417612799

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.426 × 10⁹⁸(99-digit number)

34264827329559882478…75126927560417612801

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

6.852 × 10⁹⁸(99-digit number)

68529654659119764957…50253855120835225599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.852 × 10⁹⁸(99-digit number)

68529654659119764957…50253855120835225601

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

1.370 × 10⁹⁹(100-digit number)

13705930931823952991…00507710241670451199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.370 × 10⁹⁹(100-digit number)

13705930931823952991…00507710241670451201

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

2.741 × 10⁹⁹(100-digit number)

27411861863647905983…01015420483340902399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.741 × 10⁹⁹(100-digit number)

27411861863647905983…01015420483340902401

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

5.482 × 10⁹⁹(100-digit number)

54823723727295811966…02030840966681804799

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