Block #213,846

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 10/17/2013, 2:48:01 AM · Difficulty 9.9226 · 6,578,896 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

db265eec3b8d2c54f627c9eb815e222880f9d896c1fdf5b16f0877dbdd59da0f

View on Chainz ↗

Height

#213,846

Difficulty

9.922636

Transactions

Size

200 B

Version

Bits

09ec31e6

Nonce

10,994

Timestamp

10/17/2013, 2:48:01 AM

Confirmations

6,578,896

Merkle Root

dfed9dee90cd2c2e2218c4bfceeca42e59436a0d8fed66f74198f73eb3839831

Transactions (1)

Coinbasedfed9dee90cd2c…98f73eb3839831

1 in → 1 out10.1400 XPM109 B

AWsEijaA…9WbDL6tH

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.

8.315 × 10⁹⁶(97-digit number)

83153024053563410682…15541053527748401919

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

8.315 × 10⁹⁶(97-digit number)

83153024053563410682…15541053527748401919

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.315 × 10⁹⁶(97-digit number)

83153024053563410682…15541053527748401921

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

1.663 × 10⁹⁷(98-digit number)

16630604810712682136…31082107055496803839

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.663 × 10⁹⁷(98-digit number)

16630604810712682136…31082107055496803841

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

3.326 × 10⁹⁷(98-digit number)

33261209621425364272…62164214110993607679

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.326 × 10⁹⁷(98-digit number)

33261209621425364272…62164214110993607681

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

6.652 × 10⁹⁷(98-digit number)

66522419242850728545…24328428221987215359

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.652 × 10⁹⁷(98-digit number)

66522419242850728545…24328428221987215361

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

1.330 × 10⁹⁸(99-digit number)

13304483848570145709…48656856443974430719

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.330 × 10⁹⁸(99-digit number)

13304483848570145709…48656856443974430721

Verify on FactorDB ↗Wolfram Alpha ↗

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

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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