Block #414,207

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/21/2014, 7:47:22 PM · Difficulty 10.4084 · 6,395,915 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

08d1a1195087acc9a64e412df5395fe474095f03c51f37bad3910e8e969155f2

View on Chainz ↗

Height

#414,207

Difficulty

10.408355

Transactions

Size

4.18 KB

Version

Bits

0a6889f2

Nonce

65,507

Timestamp

2/21/2014, 7:47:22 PM

Confirmations

6,395,915

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

a8d519209b728d461ccc4999d26b5a77fffee43d7b240079d9d1c4ae7703264c

Transactions (3)

Coinbase9c3f88675429d5…397193e2d4dce9

1 in → 1 out9.2700 XPM110 B

AeKNrjNj…1NEq95Ta

86f2062be897d0…79219bf33d5515

19 in → 45 out170.0511 XPM3.65 KB

AGG7FeSC…hnzYv7mQ AHb1a8RD…pF5zNFPM Ab43x36J…9xTx3Ehr AahpcZUi…qkuqDHbZ AKTn53bX…U1L287cN

b905a281bb9ffc…623c9e577dca0b

2 in → 1 out0.0341 XPM341 B

AGhWRYsP…LvRbrEq3

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.

9.014 × 10¹⁰⁷(108-digit number)

90141130190188608975…54563017224525132799

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

9.014 × 10¹⁰⁷(108-digit number)

90141130190188608975…54563017224525132799

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

9.014 × 10¹⁰⁷(108-digit number)

90141130190188608975…54563017224525132801

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

1.802 × 10¹⁰⁸(109-digit number)

18028226038037721795…09126034449050265599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.802 × 10¹⁰⁸(109-digit number)

18028226038037721795…09126034449050265601

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

3.605 × 10¹⁰⁸(109-digit number)

36056452076075443590…18252068898100531199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.605 × 10¹⁰⁸(109-digit number)

36056452076075443590…18252068898100531201

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

7.211 × 10¹⁰⁸(109-digit number)

72112904152150887180…36504137796201062399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.211 × 10¹⁰⁸(109-digit number)

72112904152150887180…36504137796201062401

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

1.442 × 10¹⁰⁹(110-digit number)

14422580830430177436…73008275592402124799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.442 × 10¹⁰⁹(110-digit number)

14422580830430177436…73008275592402124801

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 #414,207

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