Block #414,863

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/22/2014, 8:05:53 AM · Difficulty 10.3988 · 6,391,927 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f542a8952f772ebc1e1f9e5a6a4b4c1977c9c572c157998d0edcf39ca72aa3a8

View on Chainz ↗

Height

#414,863

Difficulty

10.398806

Transactions

Size

8.61 KB

Version

Bits

0a66182a

Nonce

18,421

Timestamp

2/22/2014, 8:05:53 AM

Confirmations

6,391,927

Mined by

⛏️ P2SHAbwWkWZtvvKWFxiUbZcGD749fwkawDhufa

Merkle Root

465680f5e325b70fd6ed6c77179cb2ec1bad93a59f0876a86b7a26f24577df0c

Transactions (3)

Coinbase9790c591cf01f0…bd43c77e0f98cd

1 in → 1 out9.3300 XPM116 B

AbwWkWZt…kawDhufa

112732aa7b47eb…ca5823cc31c153

65 in → 2 out600.0101 XPM7.34 KB

AdRfXoAz…mh31AgUJ AWUYtx7D…nASC8a9G

8867073616282f…191220a576bca8

5 in → 15 out46.1100 XPM1.07 KB

ATk4yXJY…yVVTNX8F AXn1KW6P…c7VDu2zv Ae5BfFib…nW1Bzjon ANGAxNk6…Bb4uN2Ub AJCY4vXj…F6vHctZE

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.

8.267 × 10⁹⁷(98-digit number)

82676685481215178599…81647032217200074239

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

8.267 × 10⁹⁷(98-digit number)

82676685481215178599…81647032217200074239

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.267 × 10⁹⁷(98-digit number)

82676685481215178599…81647032217200074241

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

1.653 × 10⁹⁸(99-digit number)

16535337096243035719…63294064434400148479

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.653 × 10⁹⁸(99-digit number)

16535337096243035719…63294064434400148481

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

3.307 × 10⁹⁸(99-digit number)

33070674192486071439…26588128868800296959

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.307 × 10⁹⁸(99-digit number)

33070674192486071439…26588128868800296961

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

6.614 × 10⁹⁸(99-digit number)

66141348384972142879…53176257737600593919

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.614 × 10⁹⁸(99-digit number)

66141348384972142879…53176257737600593921

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

1.322 × 10⁹⁹(100-digit number)

13228269676994428575…06352515475201187839

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.322 × 10⁹⁹(100-digit number)

13228269676994428575…06352515475201187841

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 #414,863

Cookie Preferences