Block #265,105

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/19/2013, 6:46:46 AM · Difficulty 9.9632 · 6,580,023 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

3e26136b81a3c6187f7c8b0cd9f54a62fb187f7899437de276f94abd2bbc0526

View on Chainz ↗

Height

#265,105

Difficulty

9.963217

Transactions

Size

455 B

Version

Bits

09f69569

Nonce

17,432

Timestamp

11/19/2013, 6:46:46 AM

Confirmations

6,580,023

Mined by

⛏️ P2SHAKcbk49N7DP3nse7MJr3Ed6rZiGJbKa7j1

Merkle Root

0c1f53dde0de98d3eb1eced1816cf3e41bab48e35762d6fb9aaff69fac5bfba0

Transactions (2)

Coinbase28237af8e46264…15884e5bfc54e6

1 in → 2 out10.0700 XPM137 B

AKcbk49N…GJbKa7j1 AU6kq4CL…dQBLvxLp

64d950fc2b0e13…ea638592803a20

1 in → 2 out1.9800 XPM227 B

AZqEHrSg…kVsvXn3C ANoad7j2…ZG9RK1pD

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.

1.694 × 10⁹⁶(97-digit number)

16945361356881986756…01539866452674198059

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

1.694 × 10⁹⁶(97-digit number)

16945361356881986756…01539866452674198059

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.694 × 10⁹⁶(97-digit number)

16945361356881986756…01539866452674198061

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

3.389 × 10⁹⁶(97-digit number)

33890722713763973512…03079732905348396119

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.389 × 10⁹⁶(97-digit number)

33890722713763973512…03079732905348396121

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

6.778 × 10⁹⁶(97-digit number)

67781445427527947024…06159465810696792239

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.778 × 10⁹⁶(97-digit number)

67781445427527947024…06159465810696792241

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

1.355 × 10⁹⁷(98-digit number)

13556289085505589404…12318931621393584479

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.355 × 10⁹⁷(98-digit number)

13556289085505589404…12318931621393584481

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

2.711 × 10⁹⁷(98-digit number)

27112578171011178809…24637863242787168959

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 #265,105

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