Block #264,032

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/18/2013, 8:30:46 AM · Difficulty 9.9651 · 6,553,056 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

72083c612a70a2125960c2ab7d10f3c8eeb872b2d22fe1f3b443f0ed59fe7dfb

View on Chainz ↗

Height

#264,032

Difficulty

9.965073

Transactions

Size

1.97 KB

Version

Bits

09f70f0e

Nonce

28,207

Timestamp

11/18/2013, 8:30:46 AM

Confirmations

6,553,056

Mined by

⛏️ P2SHAKcbk49N7DP3nse7MJr3Ed6rZiGJbKa7j1

Merkle Root

8de1181921169a64765240d719d663dea8a4a7a85d205f8030f582dc8b33a1c8

Transactions (5)

Coinbase3bed0ab9fabce3…4bf5c71eb60c40

1 in → 2 out10.1000 XPM137 B

AKcbk49N…GJbKa7j1 Abiw8n3q…ZnZfgvQW

6f7d7c6a146d4e…565f96b99c2a26

2 in → 2 out3.0324 XPM375 B

AXmex2gr…uXAJRdci AGxEymbS…6U3GR3ry

0a65313bcde77f…132f23f6359ea0

2 in → 2 out3.0500 XPM374 B

AFnfjGaj…jFrMRU98 Aa3m4J5X…ZB7wLSbT

ec813756378d80…c8c15bdf36f001

2 in → 2 out1.9900 XPM374 B

AWpeFzWa…TVnNm3Cu AXpMTyQF…eF9QLoEH

c2703f17b06dff…b9c6cc6483586c

4 in → 2 out5.0300 XPM671 B

ARyrsFpv…imkeST2C AbJPf2bu…8absDfPV

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.

1.913 × 10⁹⁶(97-digit number)

19135021367466709090…49072773235959004259

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

1.913 × 10⁹⁶(97-digit number)

19135021367466709090…49072773235959004259

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.913 × 10⁹⁶(97-digit number)

19135021367466709090…49072773235959004261

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

3.827 × 10⁹⁶(97-digit number)

38270042734933418180…98145546471918008519

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.827 × 10⁹⁶(97-digit number)

38270042734933418180…98145546471918008521

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

7.654 × 10⁹⁶(97-digit number)

76540085469866836360…96291092943836017039

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

7.654 × 10⁹⁶(97-digit number)

76540085469866836360…96291092943836017041

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

1.530 × 10⁹⁷(98-digit number)

15308017093973367272…92582185887672034079

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.530 × 10⁹⁷(98-digit number)

15308017093973367272…92582185887672034081

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

3.061 × 10⁹⁷(98-digit number)

30616034187946734544…85164371775344068159

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 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

CommonChain length 9

Found in most blocks. The baseline for Primecoin's proof-of-work.

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 #264,032

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