Block #416,197

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/23/2014, 6:16:38 AM · Difficulty 10.4001 · 6,410,451 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

ed8ba68db980c77adb90cfd8155546f3574e19d40f3110d4b9223a937fff9847

View on Chainz ↗

Height

#416,197

Difficulty

10.400128

Transactions

Size

1.58 KB

Version

Bits

0a666ed1

Nonce

35,962

Timestamp

2/23/2014, 6:16:38 AM

Confirmations

6,410,451

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

743d6093f4a3bcca53ac732adc1e0c6acc1def8ce4bd64640350c5dd9dde0683

Transactions (2)

Coinbase8de1a9c81e0ebd…b58a75bdabbbaf

1 in → 1 out9.2500 XPM110 B

AeKNrjNj…1NEq95Ta

162b0b5765263e…03a1eb870ae425

7 in → 16 out46.8383 XPM1.39 KB

ANiUR1Zn…eCyWZGPj AZDjinxC…QU3gZxpJ AWusGT5L…b5tk19df AKJx7ewq…cdjZ12Jp AQxJ1VVp…i5Wf3MsE

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.

5.223 × 10⁹⁷(98-digit number)

52236023539367118785…89405380675421299999

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

5.223 × 10⁹⁷(98-digit number)

52236023539367118785…89405380675421299999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.223 × 10⁹⁷(98-digit number)

52236023539367118785…89405380675421300001

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

1.044 × 10⁹⁸(99-digit number)

10447204707873423757…78810761350842599999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.044 × 10⁹⁸(99-digit number)

10447204707873423757…78810761350842600001

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

2.089 × 10⁹⁸(99-digit number)

20894409415746847514…57621522701685199999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.089 × 10⁹⁸(99-digit number)

20894409415746847514…57621522701685200001

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

4.178 × 10⁹⁸(99-digit number)

41788818831493695028…15243045403370399999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.178 × 10⁹⁸(99-digit number)

41788818831493695028…15243045403370400001

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

8.357 × 10⁹⁸(99-digit number)

83577637662987390057…30486090806740799999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

8.357 × 10⁹⁸(99-digit number)

83577637662987390057…30486090806740800001

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 #416,197

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