Block #8,413

TWNLength 7★☆☆☆☆

Bi-Twin Chain · Discovered 7/10/2013, 4:25:58 PM · Difficulty 7.5675 · 6,806,506 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

3fa0f751235752860cdb264a3b1ef01267dcfa16e8d0e3e214f93bf81e6cd50e

View on Chainz ↗

Height

#8,413

Difficulty

7.567451

Transactions

Size

1.51 KB

Version

Bits

0791447e

Nonce

Timestamp

7/10/2013, 4:25:58 PM

Confirmations

6,806,506

Mined by

⛏️ P2SHAHd1Q2ZqjfHUXXv72hL7FRZxmVgLmthUyS

Merkle Root

570e05cb9faec75dce4519aa8efda21c43ccd65ffd2344384c81dd07093128b7

Transactions (4)

Coinbase56027b23b5e12b…bc6125870d34f1

1 in → 1 out17.4700 XPM109 B

AHd1Q2Zq…gLmthUyS

40f91acdca20be…8595b942a12606

6 in → 2 out113.7500 XPM761 B

AGC72fWp…sv4Ff7Wa AREN2GsF…iGzAjCx7

12003201a8158b…f77a19dab5f77b

1 in → 1 out18.3900 XPM157 B

AXydpnfD…YJwueXut

862fbedac0d36b…7580379f4e2777

1 in → 8 out16.4200 XPM429 B

AJFYpPaU…hMTfyLnF AMFGWw9y…MC6yHLFA AJmAMs6q…Z2h1sy4a AbCVfZAW…7DPatys5 AVy4PwsS…AbjpW3G4

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.

6.393 × 10¹⁰⁶(107-digit number)

63931274655055069537…64389424247633327949

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

6.393 × 10¹⁰⁶(107-digit number)

63931274655055069537…64389424247633327949

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.393 × 10¹⁰⁶(107-digit number)

63931274655055069537…64389424247633327951

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

1.278 × 10¹⁰⁷(108-digit number)

12786254931011013907…28778848495266655899

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.278 × 10¹⁰⁷(108-digit number)

12786254931011013907…28778848495266655901

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

2.557 × 10¹⁰⁷(108-digit number)

25572509862022027814…57557696990533311799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.557 × 10¹⁰⁷(108-digit number)

25572509862022027814…57557696990533311801

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

5.114 × 10¹⁰⁷(108-digit number)

51145019724044055629…15115393981066623599

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 7 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 7

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 #8,413

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