Block #31,936

TWNLength 7★☆☆☆☆

Bi-Twin Chain · Discovered 7/14/2013, 1:14:41 AM · Difficulty 7.9895 · 6,763,782 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

2154392574390027d9cf6cfc2c9fad188b3995f1cdd5f83fe4dec425bff15118

View on Chainz ↗

Height

#31,936

Difficulty

7.989546

Transactions

Size

198 B

Version

Bits

07fd52dd

Nonce

415

Timestamp

7/14/2013, 1:14:41 AM

Confirmations

6,763,782

Mined by

⛏️ P2SHAGemJXng9Py5jf5TzegELEhNUUw1K3MEut

Merkle Root

01f2749d2eb2faab9d74e8dfc1a55af63b594d1dcb4a0bc7aa66e1c2819ef33f

Transactions (1)

Coinbase01f2749d2eb2fa…66e1c2819ef33f

1 in → 1 out15.6500 XPM108 B

AGemJXng…w1K3MEut

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.

5.546 × 10⁹⁴(95-digit number)

55463772224024908988…22443824157592120479

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

5.546 × 10⁹⁴(95-digit number)

55463772224024908988…22443824157592120479

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.546 × 10⁹⁴(95-digit number)

55463772224024908988…22443824157592120481

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

1.109 × 10⁹⁵(96-digit number)

11092754444804981797…44887648315184240959

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.109 × 10⁹⁵(96-digit number)

11092754444804981797…44887648315184240961

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

2.218 × 10⁹⁵(96-digit number)

22185508889609963595…89775296630368481919

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.218 × 10⁹⁵(96-digit number)

22185508889609963595…89775296630368481921

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

4.437 × 10⁹⁵(96-digit number)

44371017779219927190…79550593260736963839

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 7 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 7

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