Block #1,413,286

TWNLength 12★★★★☆

Bi-Twin Chain · Discovered 1/14/2016, 4:54:58 PM · Difficulty 10.8045 · 5,411,494 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

dd6f64a2933414a97b0213e7303d56069ecbe9a69829cb10c438296053327d9a

View on Chainz ↗

Height

#1,413,286

Difficulty

10.804524

Transactions

Size

1.08 KB

Version

Bits

0acdf541

Nonce

673,418,140

Timestamp

1/14/2016, 4:54:58 PM

Confirmations

5,411,494

Mined by

⛏️ P2SHAbCFpibX7eUDQKpjz8Tktre18mDMrtA4fs

Merkle Root

20024e126c123c0d8f2d3b3e21c15f4f9a2ccefc8481c5f942607ed753d58ce3

Transactions (2)

Coinbase57df618ddd3c46…5ece7df8fbfd79

1 in → 1 out8.5600 XPM109 B

AbCFpibX…DMrtA4fs

cefee170dfdb85…e6a5b480bdfd34

1 in → 22 out10.9970 XPM906 B

Aa8v2ZHr…TefvwHiL AN8Ksnp9…FTuZ6GLW ATEte6QE…Azs9VFkt AKS8rGVa…14QWTSvS Abpwfp3B…si9a6ff2

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.

4.315 × 10⁹⁶(97-digit number)

43155700198841255469…36669600012430796799

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

4.315 × 10⁹⁶(97-digit number)

43155700198841255469…36669600012430796799

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.315 × 10⁹⁶(97-digit number)

43155700198841255469…36669600012430796801

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

8.631 × 10⁹⁶(97-digit number)

86311400397682510939…73339200024861593599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.631 × 10⁹⁶(97-digit number)

86311400397682510939…73339200024861593601

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

1.726 × 10⁹⁷(98-digit number)

17262280079536502187…46678400049723187199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.726 × 10⁹⁷(98-digit number)

17262280079536502187…46678400049723187201

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

3.452 × 10⁹⁷(98-digit number)

34524560159073004375…93356800099446374399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.452 × 10⁹⁷(98-digit number)

34524560159073004375…93356800099446374401

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

6.904 × 10⁹⁷(98-digit number)

69049120318146008751…86713600198892748799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

6.904 × 10⁹⁷(98-digit number)

69049120318146008751…86713600198892748801

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

Level 5 — Twin Prime Pair (2^5 × origin ± 1)

2^5 × origin − 1

1.380 × 10⁹⁸(99-digit number)

13809824063629201750…73427200397785497599

Verify on FactorDB ↗Wolfram Alpha ↗

2^5 × origin + 1

1.380 × 10⁹⁸(99-digit number)

13809824063629201750…73427200397785497601

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^5 × origin + 1 − 2^5 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 12 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

ExceptionalChain length 12

Around 1 in 10,000 blocks. A significant mathematical achievement.

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 #1,413,286

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