Block #419,429

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/25/2014, 4:31:48 PM · Difficulty 10.3785 · 6,391,288 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

32be88a5cf53604507d98f440b40d7260b8a6d1a5855a9f2a58a40f050683c53

View on Chainz ↗

Height

#419,429

Difficulty

10.378463

Transactions

Size

881 B

Version

Bits

0a60e2f5

Nonce

48,829

Timestamp

2/25/2014, 4:31:48 PM

Confirmations

6,391,288

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

932fecc6560710e4f1c082fc8d2bfedfce53c867b4e50eb0725ad294f5fd235d

Transactions (3)

Coinbase8dc13ced572be3…47058a36d79272

1 in → 1 out9.2907 XPM110 B

AeKNrjNj…1NEq95Ta

bbe7185cac21b3…fefa2a5857771a

3 in → 1 out14.3000 XPM487 B

ATR6Ygr1…BYtqS7Cs

c6ae74ea8ce695…b56e3a5bdee0e1

1 in → 1 out2.9900 XPM192 B

AUvmGj9D…fchFX8bk

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.

1.681 × 10⁹⁹(100-digit number)

16816247445303159971…75926731365178816479

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

1.681 × 10⁹⁹(100-digit number)

16816247445303159971…75926731365178816479

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.681 × 10⁹⁹(100-digit number)

16816247445303159971…75926731365178816481

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

3.363 × 10⁹⁹(100-digit number)

33632494890606319942…51853462730357632959

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.363 × 10⁹⁹(100-digit number)

33632494890606319942…51853462730357632961

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

6.726 × 10⁹⁹(100-digit number)

67264989781212639884…03706925460715265919

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.726 × 10⁹⁹(100-digit number)

67264989781212639884…03706925460715265921

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

1.345 × 10¹⁰⁰(101-digit number)

13452997956242527976…07413850921430531839

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.345 × 10¹⁰⁰(101-digit number)

13452997956242527976…07413850921430531841

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

2.690 × 10¹⁰⁰(101-digit number)

26905995912485055953…14827701842861063679

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.690 × 10¹⁰⁰(101-digit number)

26905995912485055953…14827701842861063681

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 #419,429

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