Block #163,405

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/13/2013, 11:32:48 PM · Difficulty 9.8618 · 6,629,290 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

8d582e0b90a894adfe918b7e1768904331b2468744ee8d674cc87efa6960981b

View on Chainz ↗

Height

#163,405

Difficulty

9.861787

Transactions

Size

198 B

Version

Bits

09dc9e15

Nonce

123,697

Timestamp

9/13/2013, 11:32:48 PM

Confirmations

6,629,290

Merkle Root

608937dc3ab85bed2032102101e7f6c5bcbe239272f01c178926c7b28d0ff481

Transactions (1)

Coinbase608937dc3ab85b…26c7b28d0ff481

1 in → 1 out10.2700 XPM109 B

AH8MuWec…2bVG62mK

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.

9.942 × 10⁹¹(92-digit number)

99428724594446500468…19792856554228224939

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

9.942 × 10⁹¹(92-digit number)

99428724594446500468…19792856554228224939

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

9.942 × 10⁹¹(92-digit number)

99428724594446500468…19792856554228224941

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

1.988 × 10⁹²(93-digit number)

19885744918889300093…39585713108456449879

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.988 × 10⁹²(93-digit number)

19885744918889300093…39585713108456449881

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

3.977 × 10⁹²(93-digit number)

39771489837778600187…79171426216912899759

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.977 × 10⁹²(93-digit number)

39771489837778600187…79171426216912899761

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

7.954 × 10⁹²(93-digit number)

79542979675557200375…58342852433825799519

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.954 × 10⁹²(93-digit number)

79542979675557200375…58342852433825799521

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

1.590 × 10⁹³(94-digit number)

15908595935111440075…16685704867651599039

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