Block #291,045

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 12/2/2013, 11:54:01 PM · Difficulty 9.9896 · 6,534,269 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

de39c0be3a9d9c708de8ed8c8b9f329f9ebe4297708eadb2a4160660f90ca410

View on Chainz ↗

Height

#291,045

Difficulty

9.989619

Transactions

Size

1.00 KB

Version

Bits

09fd57b3

Nonce

2,771

Timestamp

12/2/2013, 11:54:01 PM

Confirmations

6,534,269

Mined by

⛏️ paRAQwRN8bEjE7sawFBq7N38V9niAV8PsTcY9

Merkle Root

a7a4c5e3e54dad7662d613a0ea6d3809877f3b653979a09df692a7c2923ebe18

Transactions (2)

Coinbase3015e897a75090…0171a1fde10cf6

1 in → 19 out10.0100 XPM709 B

AQwRN8bE…V8PsTcY9 AGPYJBwg…5yWMRKmU ASdxUYDX…6mVReCnB Ad9pSzHt…cpJ3y2zz AayReikY…2HAC42fs

666cf0e934a405…e449d68cc2a64d

1 in → 2 out2.5225 XPM226 B

AUxfou8w…KAAGRuM1 APWKxwCK…wU1BQKNQ

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.

1.684 × 10⁹¹(92-digit number)

16841738271229535509…37323433891860532499

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

1.684 × 10⁹¹(92-digit number)

16841738271229535509…37323433891860532499

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.684 × 10⁹¹(92-digit number)

16841738271229535509…37323433891860532501

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

3.368 × 10⁹¹(92-digit number)

33683476542459071019…74646867783721064999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.368 × 10⁹¹(92-digit number)

33683476542459071019…74646867783721065001

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

6.736 × 10⁹¹(92-digit number)

67366953084918142038…49293735567442129999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.736 × 10⁹¹(92-digit number)

67366953084918142038…49293735567442130001

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

1.347 × 10⁹²(93-digit number)

13473390616983628407…98587471134884259999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.347 × 10⁹²(93-digit number)

13473390616983628407…98587471134884260001

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

2.694 × 10⁹²(93-digit number)

26946781233967256815…97174942269768519999

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 #291,045

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