Block #407,438

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/16/2014, 11:07:11 PM · Difficulty 10.4321 · 6,385,226 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e948c9fe33d28c8b8054b09a0b5541d61f9ca4b26a449bc804a3feb85fc28a74

View on Chainz ↗

Height

#407,438

Difficulty

10.432063

Transactions

Size

1.54 KB

Version

Bits

0a6e9bb0

Nonce

62,220

Timestamp

2/16/2014, 11:07:11 PM

Confirmations

6,385,226

Merkle Root

5d63aad5d22c1d4f534359ff8e6429676013db558ff590fc3efb223cafcf7d08

Transactions (5)

Coinbase803d620629d26d…dde90a9c37b868

1 in → 1 out9.2100 XPM109 B

AMFcvyqN…8VeZ11QK

815615df675ff8…bafdc1cdb56a75

4 in → 2 out25.2214 XPM670 B

Abxn3EEB…8ccf7zVj AWK6bMgW…Ae6khPuN

62667708da4864…3359ae2cca8d3b

1 in → 4 out9.2100 XPM260 B

AMBGfmzX…Q27ynGeT AKc9Rmjx…Bir3N5mm AeKNrjNj…1NEq95Ta ALvVMKLt…wbxp4TJH

7901214528cf7b…c4e788a2938c90

1 in → 2 out5.0845 XPM225 B

ANWDkUqV…MU5m98J4 AbnfpZ84…Qu96zKV4

18c90fa4c047e9…fa4091a0f6d996

1 in → 2 out4.0879 XPM226 B

AY93eJkF…8yAjzHJe Abkp6z5P…hQ5DLwan

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.

5.653 × 10⁹⁸(99-digit number)

56532935418862721338…28595418980296548399

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

5.653 × 10⁹⁸(99-digit number)

56532935418862721338…28595418980296548399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.653 × 10⁹⁸(99-digit number)

56532935418862721338…28595418980296548401

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

1.130 × 10⁹⁹(100-digit number)

11306587083772544267…57190837960593096799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.130 × 10⁹⁹(100-digit number)

11306587083772544267…57190837960593096801

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

2.261 × 10⁹⁹(100-digit number)

22613174167545088535…14381675921186193599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.261 × 10⁹⁹(100-digit number)

22613174167545088535…14381675921186193601

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

4.522 × 10⁹⁹(100-digit number)

45226348335090177070…28763351842372387199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.522 × 10⁹⁹(100-digit number)

45226348335090177070…28763351842372387201

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

9.045 × 10⁹⁹(100-digit number)

90452696670180354141…57526703684744774399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

9.045 × 10⁹⁹(100-digit number)

90452696670180354141…57526703684744774401

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