Block #291,293

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/3/2013, 2:36:20 AM · Difficulty 9.9898 · 6,519,582 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

34b4fc0a108c16b2e4d6930afbd18c28c353eb53d6392c45aacf2d665bf2693e

View on Chainz ↗

Height

#291,293

Difficulty

9.989780

Transactions

Size

1003 B

Version

Bits

09fd623b

Nonce

16,096

Timestamp

12/3/2013, 2:36:20 AM

Confirmations

6,519,582

Mined by

⛏️ qaRFAZninDSu1Gy3NdWkaTGfLDa2i9FvgDMwC4

Merkle Root

88e918015b4be7aa4ab0a38a26ac1fd7e3711f17cfe45a826df32023b02ca256

Transactions (1)

Coinbase88e918015b4be7…f32023b02ca256

1 in → 25 out10.0081 XPM913 B

AZninDSu…FvgDMwC4 Ac5sZcdP…kzV4eckG APFsQ3Fo…xoFKrF12 Ae58nAEb…GFUA15fv AKde1i4d…88R1bw6g

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.

2.111 × 10⁹⁴(95-digit number)

21110472325895222193…21058070438730568599

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

2.111 × 10⁹⁴(95-digit number)

21110472325895222193…21058070438730568599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.111 × 10⁹⁴(95-digit number)

21110472325895222193…21058070438730568601

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

4.222 × 10⁹⁴(95-digit number)

42220944651790444386…42116140877461137199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.222 × 10⁹⁴(95-digit number)

42220944651790444386…42116140877461137201

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

8.444 × 10⁹⁴(95-digit number)

84441889303580888772…84232281754922274399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.444 × 10⁹⁴(95-digit number)

84441889303580888772…84232281754922274401

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

1.688 × 10⁹⁵(96-digit number)

16888377860716177754…68464563509844548799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.688 × 10⁹⁵(96-digit number)

16888377860716177754…68464563509844548801

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

3.377 × 10⁹⁵(96-digit number)

33776755721432355508…36929127019689097599

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.377 × 10⁹⁵(96-digit number)

33776755721432355508…36929127019689097601

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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 #291,293

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