Block #413,597

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/21/2014, 8:42:20 AM · Difficulty 10.4142 · 6,401,412 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

97f9086165571cb7607347b971b4a131d7f6593aa7fcb34022bd29a7df9fc04e

View on Chainz ↗

Height

#413,597

Difficulty

10.414238

Transactions

Size

652 B

Version

Bits

0a6a0b7f

Nonce

100,567

Timestamp

2/21/2014, 8:42:20 AM

Confirmations

6,401,412

Mined by

⛏️ P2SHAa8wFKioJSpVwB1sZfJg1X8r2oNVKtbmya

Merkle Root

3a8c69a90c81ae86169d8192a60ff5ae4482a064a59513b7bddf67a228a7f114

Transactions (3)

Coinbase7cebc7c27c7e65…c7e45ac98ed7dc

1 in → 1 out9.2300 XPM109 B

Aa8wFKio…NVKtbmya

18d67f507901a8…185c584ea6ec68

1 in → 2 out3.1121 XPM227 B

AHkGXFRC…1qeN2Q7U AKDHtASo…gStJpR57

616392968abbab…48dc910b08aa16

1 in → 2 out1.1401 XPM225 B

AMiVTMtt…4Knq1ve9 AL7mrNmD…pu17EQF5

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.

4.650 × 10⁹⁷(98-digit number)

46500073120439854045…76244064563772044319

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

4.650 × 10⁹⁷(98-digit number)

46500073120439854045…76244064563772044319

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.650 × 10⁹⁷(98-digit number)

46500073120439854045…76244064563772044321

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

9.300 × 10⁹⁷(98-digit number)

93000146240879708091…52488129127544088639

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

9.300 × 10⁹⁷(98-digit number)

93000146240879708091…52488129127544088641

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

1.860 × 10⁹⁸(99-digit number)

18600029248175941618…04976258255088177279

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.860 × 10⁹⁸(99-digit number)

18600029248175941618…04976258255088177281

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

3.720 × 10⁹⁸(99-digit number)

37200058496351883236…09952516510176354559

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.720 × 10⁹⁸(99-digit number)

37200058496351883236…09952516510176354561

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

7.440 × 10⁹⁸(99-digit number)

74400116992703766473…19905033020352709119

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

7.440 × 10⁹⁸(99-digit number)

74400116992703766473…19905033020352709121

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

Block #413,597

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