Block #131,709

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 8/24/2013, 11:24:19 AM · Difficulty 9.7857 · 6,676,420 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

34514c302424075cb2e2f0443e1282f2cee12dc4f76f374430a7e4a7f9200d20

View on Chainz ↗

Height

#131,709

Difficulty

9.785688

Transactions

Size

1.37 KB

Version

Bits

09c922e0

Nonce

141,650

Timestamp

8/24/2013, 11:24:19 AM

Confirmations

6,676,420

Mined by

⛏️ P2SHAeGSDEJdHhg3wBrXmwG4Fazqdsmt5GEof2

Merkle Root

be91ff3bf0207a9c09e09c68174aeeb9201000b119243e0d2be0d20ce9db15fb

Transactions (4)

Coinbase2abb75f1af138b…88c910dde0344b

1 in → 1 out10.4600 XPM109 B

AeGSDEJd…mt5GEof2

2bb64aa79df095…22f7c4bb6b585c

1 in → 2 out10.4200 XPM192 B

AGZvBxxa…shhkwtTn AcigC6jS…boDz1qQv

7ea509cd226575…f57b3c2b19ed5d

1 in → 2 out10.4200 XPM192 B

AK4rDdii…1TURaAFf AGZvBxxa…shhkwtTn

985fdb3b234743…7d5997dae880df

5 in → 2 out2.1031 XPM818 B

Aa7CSG8M…SMpjerCk AWe8udVp…2BB4QFAv

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.

1.155 × 10⁹⁸(99-digit number)

11556126085872239403…09928785819647757499

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

1.155 × 10⁹⁸(99-digit number)

11556126085872239403…09928785819647757499

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.155 × 10⁹⁸(99-digit number)

11556126085872239403…09928785819647757501

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

2.311 × 10⁹⁸(99-digit number)

23112252171744478806…19857571639295514999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.311 × 10⁹⁸(99-digit number)

23112252171744478806…19857571639295515001

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

4.622 × 10⁹⁸(99-digit number)

46224504343488957613…39715143278591029999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.622 × 10⁹⁸(99-digit number)

46224504343488957613…39715143278591030001

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

9.244 × 10⁹⁸(99-digit number)

92449008686977915227…79430286557182059999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

9.244 × 10⁹⁸(99-digit number)

92449008686977915227…79430286557182060001

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

1.848 × 10⁹⁹(100-digit number)

18489801737395583045…58860573114364119999

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 #131,709

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