Block #158,213

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/10/2013, 5:07:37 AM · Difficulty 9.8678 · 6,651,958 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

930d4220648c0b77428ee3f5937dffaf8a572d428d346a2f87b62eaae1e7ea29

View on Chainz ↗

Height

#158,213

Difficulty

9.867786

Transactions

Size

1.11 KB

Version

Bits

09de273f

Nonce

97,056

Timestamp

9/10/2013, 5:07:37 AM

Confirmations

6,651,958

Mined by

⛏️ P2SHALrneVzuYAwbT7j7unDhrUBYvEMwH8VrKf

Merkle Root

ce4415c7436a8a4a1ecf8dd60ccae343bfb9872a0cad8496db32bdc6dd43f11e

Transactions (3)

Coinbase42be0de001b9dd…652be701586022

1 in → 1 out10.2700 XPM109 B

ALrneVzu…MwH8VrKf

97e81da343ae70…19fd8daaed88e1

1 in → 3 out10.2500 XPM226 B

AU5g73DE…hY8WccpK AHxxCL1z…jmPH7fKk Af55M7K2…Zq95d6gB

1a850f5085df4b…3a12e34a38917e

5 in → 2 out38.6560 XPM714 B

ARTD9mc6…EigiufJf AGZvBxxa…shhkwtTn

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.

1.478 × 10⁹⁰(91-digit number)

14786214903043005310…05300544410006936319

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

1.478 × 10⁹⁰(91-digit number)

14786214903043005310…05300544410006936319

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.478 × 10⁹⁰(91-digit number)

14786214903043005310…05300544410006936321

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

2.957 × 10⁹⁰(91-digit number)

29572429806086010621…10601088820013872639

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.957 × 10⁹⁰(91-digit number)

29572429806086010621…10601088820013872641

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

5.914 × 10⁹⁰(91-digit number)

59144859612172021243…21202177640027745279

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.914 × 10⁹⁰(91-digit number)

59144859612172021243…21202177640027745281

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

1.182 × 10⁹¹(92-digit number)

11828971922434404248…42404355280055490559

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.182 × 10⁹¹(92-digit number)

11828971922434404248…42404355280055490561

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

2.365 × 10⁹¹(92-digit number)

23657943844868808497…84808710560110981119

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 #158,213

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