Block #233,698

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/29/2013, 9:14:12 PM · Difficulty 9.9430 · 6,565,660 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

7cd01e713ebe43126b6bf41dd0995ad09acdb4b47d58f07307eabe34825844d1

View on Chainz ↗

Height

#233,698

Difficulty

9.942989

Transactions

Size

1.91 KB

Version

Bits

09f167be

Nonce

75,758

Timestamp

10/29/2013, 9:14:12 PM

Confirmations

6,565,660

Mined by

⛏️ Rp%AP72ZF1f4Tt3L7vBSP2XmkGsa8K2ngR7CZ

Merkle Root

3c338c5687549a5e21ded9cec285743dc7bc801e0d81b304ff361e32253b3e24

Transactions (1)

Coinbase3c338c5687549a…361e32253b3e24

1 in → 53 out10.0800 XPM1.82 KB

AP72ZF1f…K2ngR7CZ AdwTEXyC…NwbVbqZV ANuKx9Ke…wm7nrBRq ARpndEmc…Hg792kEk AQtzUSwq…htAuF3yC

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.

1.608 × 10⁹¹(92-digit number)

16085245911355166185…37806083447694800319

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

1.608 × 10⁹¹(92-digit number)

16085245911355166185…37806083447694800319

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.608 × 10⁹¹(92-digit number)

16085245911355166185…37806083447694800321

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

3.217 × 10⁹¹(92-digit number)

32170491822710332371…75612166895389600639

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.217 × 10⁹¹(92-digit number)

32170491822710332371…75612166895389600641

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

6.434 × 10⁹¹(92-digit number)

64340983645420664742…51224333790779201279

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.434 × 10⁹¹(92-digit number)

64340983645420664742…51224333790779201281

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

1.286 × 10⁹²(93-digit number)

12868196729084132948…02448667581558402559

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.286 × 10⁹²(93-digit number)

12868196729084132948…02448667581558402561

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

2.573 × 10⁹²(93-digit number)

25736393458168265897…04897335163116805119

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