Block #133,865

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 8/25/2013, 6:24:10 PM · Difficulty 9.7980 · 6,681,045 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

d4f6650d444d6024199b95af76335b73d14c2c66f5a8aca3b68dba32e270c45e

View on Chainz ↗

Height

#133,865

Difficulty

9.798010

Transactions

Size

585 B

Version

Bits

09cc4a60

Nonce

3,936

Timestamp

8/25/2013, 6:24:10 PM

Confirmations

6,681,045

Mined by

⛏️ P2SHAbB3aifuT9ri2zqZMbJ723U3z6wrbJL6FT

Merkle Root

9a6644111840762245cd98c350751453038afe7e0f6340c5b32afc153258a3ce

Transactions (2)

Coinbase474f31d11c965d…b95614cc0244f6

1 in → 1 out10.4100 XPM109 B

AbB3aifu…wrbJL6FT

0ce74006fb9bec…0796d054472fcd

3 in → 1 out31.3400 XPM387 B

AKSrQmiZ…7NmHtCGr

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.

3.343 × 10⁹²(93-digit number)

33430598980957502622…29575482821783065199

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

3.343 × 10⁹²(93-digit number)

33430598980957502622…29575482821783065199

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.343 × 10⁹²(93-digit number)

33430598980957502622…29575482821783065201

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

6.686 × 10⁹²(93-digit number)

66861197961915005244…59150965643566130399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.686 × 10⁹²(93-digit number)

66861197961915005244…59150965643566130401

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

1.337 × 10⁹³(94-digit number)

13372239592383001048…18301931287132260799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.337 × 10⁹³(94-digit number)

13372239592383001048…18301931287132260801

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

2.674 × 10⁹³(94-digit number)

26744479184766002097…36603862574264521599

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.674 × 10⁹³(94-digit number)

26744479184766002097…36603862574264521601

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

5.348 × 10⁹³(94-digit number)

53488958369532004195…73207725148529043199

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 #133,865

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