Block #413,515

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/21/2014, 7:16:49 AM · Difficulty 10.4144 · 6,400,525 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

761c9159dfa67131dfc18ac52e59dea7245e18458be692890cc906ed0d3dbcc3

View on Chainz ↗

Height

#413,515

Difficulty

10.414354

Transactions

Size

7.00 KB

Version

Bits

0a6a131c

Nonce

102,177

Timestamp

2/21/2014, 7:16:49 AM

Confirmations

6,400,525

Mined by

⛏️ KOaSAQvZBsUoAqCaKecR1VEueWTuBchppgRZTJ

Merkle Root

9459a8d8a5c2d7f9b046ee10d8ebb2014ca09b6de31fd8a206de9c77d62e9822

Transactions (4)

Coinbase880986566f4572…942a5c156bf3cd

1 in → 21 out9.2100 XPM777 B

AQvZBsUo…hppgRZTJ AZV4TwxQ…8JnQqSoH Aac9Hjdg…P4ktypc3 AYtDvcGs…VuAcA7m7 AGKhfQo2…h7fBXLX6

c363fc93e9ae2f…4df95a4d44a66e

38 in → 2 out38.1603 XPM5.57 KB

ARu3tR3L…HgFgX9ti AQPDSwA3…wiu2tENs

4db9d3df1028b5…3984a460ac767a

2 in → 2 out0.4104 XPM375 B

APzSEqan…HYkSzq8E AVZjVdGK…dwarKxHX

7c2219f26adeff…bed6b1e9815da1

1 in → 2 out4.1140 XPM225 B

AarpnjQF…4ATdxNud Acw6nebH…8mcry7u8

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.

8.032 × 10⁹⁸(99-digit number)

80329791449965868940…01307485957632168959

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

8.032 × 10⁹⁸(99-digit number)

80329791449965868940…01307485957632168959

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.032 × 10⁹⁸(99-digit number)

80329791449965868940…01307485957632168961

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

1.606 × 10⁹⁹(100-digit number)

16065958289993173788…02614971915264337919

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.606 × 10⁹⁹(100-digit number)

16065958289993173788…02614971915264337921

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

3.213 × 10⁹⁹(100-digit number)

32131916579986347576…05229943830528675839

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.213 × 10⁹⁹(100-digit number)

32131916579986347576…05229943830528675841

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

6.426 × 10⁹⁹(100-digit number)

64263833159972695152…10459887661057351679

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.426 × 10⁹⁹(100-digit number)

64263833159972695152…10459887661057351681

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

1.285 × 10¹⁰⁰(101-digit number)

12852766631994539030…20919775322114703359

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.285 × 10¹⁰⁰(101-digit number)

12852766631994539030…20919775322114703361

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 #413,515

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