Block #416,158

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/23/2014, 5:46:20 AM · Difficulty 10.3991 · 6,382,850 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

3f2b535618098cf17aef93f91909ec71cddc56aec605f812f2266051329caeda

View on Chainz ↗

Height

#416,158

Difficulty

10.399124

Transactions

Size

1.51 KB

Version

Bits

0a662d00

Nonce

2,019

Timestamp

2/23/2014, 5:46:20 AM

Confirmations

6,382,850

Mined by

⛏️ YaSAQvZBsUoAqCaKecR1VEueWTuBchppgRZTJ

Merkle Root

677dbdd632c7146123abe50c03d94723416c10052c8dc218fa2f39694d1f3136

Transactions (4)

Coinbasef844291c5a39c8…abd9ef7f567c57

1 in → 21 out9.2300 XPM777 B

AQvZBsUo…hppgRZTJ AZV4TwxQ…8JnQqSoH Aac9Hjdg…P4ktypc3 AGKhfQo2…h7fBXLX6 ALqxX1WA…hsKjbnXn

320efe6b01dd5d…782fa3331e42e4

1 in → 2 out9.1800 XPM226 B

ARrcZB2q…nLiz6Sz4 AeYkZ5US…aRET5FTi

c527e546f6fd8c…8f550d6b7d1913

1 in → 2 out9.1800 XPM226 B

AWpw2ZoJ…1piCs7qb APyCJcr1…QAhukZCb

335786ecff144f…88490c183611b7

1 in → 2 out9.1800 XPM227 B

AZNE54tz…Mi3QSvq6 AMWhfZow…EH5vsb34

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.

2.620 × 10⁹⁷(98-digit number)

26208496929510125307…47379495788927077119

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

2.620 × 10⁹⁷(98-digit number)

26208496929510125307…47379495788927077119

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.620 × 10⁹⁷(98-digit number)

26208496929510125307…47379495788927077121

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

5.241 × 10⁹⁷(98-digit number)

52416993859020250614…94758991577854154239

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.241 × 10⁹⁷(98-digit number)

52416993859020250614…94758991577854154241

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

1.048 × 10⁹⁸(99-digit number)

10483398771804050122…89517983155708308479

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.048 × 10⁹⁸(99-digit number)

10483398771804050122…89517983155708308481

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

2.096 × 10⁹⁸(99-digit number)

20966797543608100245…79035966311416616959

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.096 × 10⁹⁸(99-digit number)

20966797543608100245…79035966311416616961

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

4.193 × 10⁹⁸(99-digit number)

41933595087216200491…58071932622833233919

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.193 × 10⁹⁸(99-digit number)

41933595087216200491…58071932622833233921

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