Block #3,865

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/9/2013, 8:08:52 AM · Difficulty 7.2735 · 6,785,667 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

ff5a9b8552ae4c6fa409d11a3308c589344487e2b4c81b8a4162b082ca5935e6

View on Chainz ↗

Height

#3,865

Difficulty

7.273510

Transactions

Size

211 B

Version

Bits

074604bb

Nonce

Timestamp

7/9/2013, 8:08:52 AM

Confirmations

6,785,667

Merkle Root

20340f5839af172f59ec0ecdba06526d873ba6da464510a0be1b75a074e6355c

Transactions (1)

Coinbase20340f5839af17…1b75a074e6355c

1 in → 1 out18.8800 XPM108 B

AP6a11Hi…J5qsJ2it

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.

6.187 × 10¹²⁵(126-digit number)

61878630894263257063…82215294416781747419

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

6.187 × 10¹²⁵(126-digit number)

61878630894263257063…82215294416781747419

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.187 × 10¹²⁵(126-digit number)

61878630894263257063…82215294416781747421

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

1.237 × 10¹²⁶(127-digit number)

12375726178852651412…64430588833563494839

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.237 × 10¹²⁶(127-digit number)

12375726178852651412…64430588833563494841

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

2.475 × 10¹²⁶(127-digit number)

24751452357705302825…28861177667126989679

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.475 × 10¹²⁶(127-digit number)

24751452357705302825…28861177667126989681

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

4.950 × 10¹²⁶(127-digit number)

49502904715410605650…57722355334253979359

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.950 × 10¹²⁶(127-digit number)

49502904715410605650…57722355334253979361

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 8 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 8

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