Block #186,058

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/29/2013, 4:48:21 PM · Difficulty 9.8640 · 6,620,207 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

a87e0bb14ac58568751c0370a2b5860c03456739e5b47d80da0fa667af53b7f4

View on Chainz ↗

Height

#186,058

Difficulty

9.864033

Transactions

Size

1.00 KB

Version

Bits

09dd313e

Nonce

11,751

Timestamp

9/29/2013, 4:48:21 PM

Confirmations

6,620,207

Mined by

⛏️ P2SHAagAeVyxpaLs7gprzKqRFtpVvMws8j5juJ

Merkle Root

2bae8f1caf3c7cf62ad2272c5b9efe98f1f6b7ccc9a614d9a74a897ac8e27443

Transactions (4)

Coinbasea4e4a807aa580e…77b9d5af6c162f

1 in → 1 out10.2900 XPM109 B

AagAeVyx…ws8j5juJ

9432f3c5974784…a523a828b340e4

1 in → 2 out8.1402 XPM226 B

Acj3MyoT…xdHHXRdM AMRnMSro…uq9bMMmU

4cff30bfdb6983…a3eca79ad285e3

1 in → 2 out8.1427 XPM227 B

AQAVUadb…vRTJXEze APXuvjkg…PXtRYGMK

765219fa20c8fd…9500a5ab04eb2f

2 in → 2 out2.0494 XPM373 B

AdQY1TJG…B8zunBwa AJv5WoqT…tqCRQqFk

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.

8.699 × 10⁹⁶(97-digit number)

86992207394243174290…63219799785439791999

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

8.699 × 10⁹⁶(97-digit number)

86992207394243174290…63219799785439791999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.699 × 10⁹⁶(97-digit number)

86992207394243174290…63219799785439792001

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

1.739 × 10⁹⁷(98-digit number)

17398441478848634858…26439599570879583999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.739 × 10⁹⁷(98-digit number)

17398441478848634858…26439599570879584001

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

3.479 × 10⁹⁷(98-digit number)

34796882957697269716…52879199141759167999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.479 × 10⁹⁷(98-digit number)

34796882957697269716…52879199141759168001

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

6.959 × 10⁹⁷(98-digit number)

69593765915394539432…05758398283518335999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.959 × 10⁹⁷(98-digit number)

69593765915394539432…05758398283518336001

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

1.391 × 10⁹⁸(99-digit number)

13918753183078907886…11516796567036671999

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 #186,058

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