Block #251,031

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 11/8/2013, 7:32:16 PM · Difficulty 9.9693 · 6,548,284 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

01074ba4431cd00d7fe94ff6e3424f9a0db6bbd221335fad1fa8c902f7d416de

View on Chainz ↗

Height

#251,031

Difficulty

9.969324

Transactions

Size

865 B

Version

Bits

09f825a3

Nonce

89,890

Timestamp

11/8/2013, 7:32:16 PM

Confirmations

6,548,284

Mined by

⛏️ aR};AdWEcK3rEzi97mdrbQ1dKgt3GmHNaeWgVb

Merkle Root

43c180d221171c23d7f59be246d12cdd056ad8434a8272c5ef51d915755198ba

Transactions (1)

Coinbase43c180d221171c…51d915755198ba

1 in → 21 out10.0500 XPM777 B

AdWEcK3r…HNaeWgVb APVGazMT…cDCk2nwA AJzWx2VD…j4ZSMit5 AeZuurfR…BADULmp6 ARskhKeb…UgDvvX9P

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.

4.005 × 10⁹⁰(91-digit number)

40052902326380343090…60932135995458610999

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

4.005 × 10⁹⁰(91-digit number)

40052902326380343090…60932135995458610999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.005 × 10⁹⁰(91-digit number)

40052902326380343090…60932135995458611001

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

8.010 × 10⁹⁰(91-digit number)

80105804652760686181…21864271990917221999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.010 × 10⁹⁰(91-digit number)

80105804652760686181…21864271990917222001

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

1.602 × 10⁹¹(92-digit number)

16021160930552137236…43728543981834443999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.602 × 10⁹¹(92-digit number)

16021160930552137236…43728543981834444001

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

3.204 × 10⁹¹(92-digit number)

32042321861104274472…87457087963668887999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.204 × 10⁹¹(92-digit number)

32042321861104274472…87457087963668888001

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

6.408 × 10⁹¹(92-digit number)

64084643722208548945…74914175927337775999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

6.408 × 10⁹¹(92-digit number)

64084643722208548945…74914175927337776001

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