Block #409,037

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/18/2014, 3:02:01 AM · Difficulty 10.4239 · 6,400,581 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

61c417f30620026b6b831e62f930502a215bde7c61c357875b678676e2c98987

View on Chainz ↗

Height

#409,037

Difficulty

10.423878

Transactions

Size

1.05 KB

Version

Bits

0a6c834a

Nonce

246,107

Timestamp

2/18/2014, 3:02:01 AM

Confirmations

6,400,581

Mined by

⛏️ P2SHAbwWkWZtvvKWFxiUbZcGD749fwkawDhufa

Merkle Root

6ccf58bda084bdd45844cfdbae234eec9c017c82f9d82a8092a4342dec50ed0d

Transactions (5)

Coinbase2fa2f5e7b004f2…656d0a7dbf6e8f

1 in → 1 out9.2300 XPM116 B

AbwWkWZt…kawDhufa

b444631f8d53b3…2c1a067cf9bac6

1 in → 1 out0.2932 XPM190 B

Af3a93o1…wDBHzJed

092f7138d23732…306f73cf9d2124

1 in → 2 out7.1356 XPM226 B

AUSNM44f…EqbaZ78J AJBNceXW…gt6AFX6G

b06b91086a0118…6a4064daff6147

1 in → 2 out1.7060 XPM226 B

APCza2xF…NYW1bK7F ATgPNRDd…HTaKJFoN

6445ffbe35ec65…b2b610f90e2dcc

1 in → 2 out6.1150 XPM227 B

AWjMy3MQ…6SDvk22A Ad7FHvXR…F1XqBLkP

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.

4.040 × 10⁹⁶(97-digit number)

40400916061911578636…85757612463908346399

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

4.040 × 10⁹⁶(97-digit number)

40400916061911578636…85757612463908346399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.040 × 10⁹⁶(97-digit number)

40400916061911578636…85757612463908346401

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

8.080 × 10⁹⁶(97-digit number)

80801832123823157272…71515224927816692799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.080 × 10⁹⁶(97-digit number)

80801832123823157272…71515224927816692801

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

1.616 × 10⁹⁷(98-digit number)

16160366424764631454…43030449855633385599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.616 × 10⁹⁷(98-digit number)

16160366424764631454…43030449855633385601

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

3.232 × 10⁹⁷(98-digit number)

32320732849529262909…86060899711266771199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.232 × 10⁹⁷(98-digit number)

32320732849529262909…86060899711266771201

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

6.464 × 10⁹⁷(98-digit number)

64641465699058525818…72121799422533542399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

6.464 × 10⁹⁷(98-digit number)

64641465699058525818…72121799422533542401

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 #409,037

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