Block #659,186

TWNLength 11★★★☆☆

Bi-Twin Chain · Discovered 8/2/2014, 10:50:48 AM · Difficulty 10.9562 · 6,135,397 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

34525309e988a8263e0b7bdadfe3e9409b56fcab83a2a1c75906d79278a04ba4

View on Chainz ↗

Height

#659,186

Difficulty

10.956223

Transactions

Size

797 B

Version

Bits

0af4cb02

Nonce

246,091

Timestamp

8/2/2014, 10:50:48 AM

Confirmations

6,135,397

Mined by

⛏️ aS.AJzWx2VDShfKP9pKK3jZqTbn4sj4ZSMit5

Merkle Root

1597a6695ea144387e3910979f9f7d49549676ecc0b98b5095bf76d95470e404

Transactions (1)

Coinbase1597a6695ea144…bf76d95470e404

1 in → 19 out8.3200 XPM708 B

AJzWx2VD…j4ZSMit5 AdRccEFQ…6E9x7zBA APfZjrNu…Wvzt6AFp AcVAEuoP…1jK61Unr AebWhrRm…K61Qrkws

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.

1.263 × 10⁹³(94-digit number)

12633433617013958015…23475644131642974079

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

1.263 × 10⁹³(94-digit number)

12633433617013958015…23475644131642974079

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.263 × 10⁹³(94-digit number)

12633433617013958015…23475644131642974081

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

2.526 × 10⁹³(94-digit number)

25266867234027916031…46951288263285948159

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.526 × 10⁹³(94-digit number)

25266867234027916031…46951288263285948161

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

5.053 × 10⁹³(94-digit number)

50533734468055832062…93902576526571896319

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.053 × 10⁹³(94-digit number)

50533734468055832062…93902576526571896321

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

1.010 × 10⁹⁴(95-digit number)

10106746893611166412…87805153053143792639

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.010 × 10⁹⁴(95-digit number)

10106746893611166412…87805153053143792641

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

2.021 × 10⁹⁴(95-digit number)

20213493787222332824…75610306106287585279

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.021 × 10⁹⁴(95-digit number)

20213493787222332824…75610306106287585281

Verify on FactorDB ↗Wolfram Alpha ↗

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

Level 5 — Twin Prime Pair (2^5 × origin ± 1)

2^5 × origin − 1

4.042 × 10⁹⁴(95-digit number)

40426987574444665649…51220612212575170559

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 11 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

RareChain length 11

Approximately 1 in 1,000 blocks. Noteworthy discoveries.

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