Block #210,135

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/14/2013, 11:06:40 PM · Difficulty 9.9125 · 6,599,357 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

61ee875d3365dbf645c9e81e27d345a9ee5d363708dbff2857a7095bdd9c754d

View on Chainz ↗

Height

#210,135

Difficulty

9.912502

Transactions

Size

4.72 KB

Version

Bits

09e999c1

Nonce

53,929

Timestamp

10/14/2013, 11:06:40 PM

Confirmations

6,599,357

Mined by

⛏️ P2SHAXkwZGZQUV996sgzkcs2jbamXUaP3jC9dV

Merkle Root

f8f395d1fa0a3fc3bf51c9ff49adce69eeb576a2d14d2da7e3b4f045f8999a09

Transactions (3)

Coinbased304d22e465f34…1d8e5a6cb0c784

1 in → 1 out10.2300 XPM109 B

AXkwZGZQ…aP3jC9dV

5d6c2fc9669042…9acf459e7db616

21 in → 2 out100.0100 XPM3.04 KB

ANfALfZo…As5bb6E4 AUWSUEty…fhtKxRFm

cc72c0a2597262…d0523d739d33f5

13 in → 1 out132.5400 XPM1.49 KB

AbxSykSg…BX71twYi

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.

1.436 × 10⁹⁹(100-digit number)

14361658926981863593…63680149299261344319

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

1.436 × 10⁹⁹(100-digit number)

14361658926981863593…63680149299261344319

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.436 × 10⁹⁹(100-digit number)

14361658926981863593…63680149299261344321

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

2.872 × 10⁹⁹(100-digit number)

28723317853963727186…27360298598522688639

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.872 × 10⁹⁹(100-digit number)

28723317853963727186…27360298598522688641

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

5.744 × 10⁹⁹(100-digit number)

57446635707927454373…54720597197045377279

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.744 × 10⁹⁹(100-digit number)

57446635707927454373…54720597197045377281

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

1.148 × 10¹⁰⁰(101-digit number)

11489327141585490874…09441194394090754559

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.148 × 10¹⁰⁰(101-digit number)

11489327141585490874…09441194394090754561

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

2.297 × 10¹⁰⁰(101-digit number)

22978654283170981749…18882388788181509119

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 #210,135

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