Block #166,156

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 9/15/2013, 6:06:35 PM · Difficulty 9.8672 · 6,637,351 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e83a3daf8d721abe3286b8cb586b5b3e5b74491862f0a0fa218ee654e60151e9

View on Chainz ↗

Height

#166,156

Difficulty

9.867156

Transactions

Size

1.06 KB

Version

Bits

09ddfdef

Nonce

45,841

Timestamp

9/15/2013, 6:06:35 PM

Confirmations

6,637,351

Mined by

⛏️ P2SHAZ8U6PTmvtjSwVPAGwxvfjNabLiCm7WwTR

Merkle Root

d2dc0cfd868a898004adb4639d51d7953d94d5df2d6711e6b92aeee9e5692905

Transactions (2)

Coinbase73bdf22e4c4f80…229b8b5b099118

1 in → 1 out10.2700 XPM109 B

AZ8U6PTm…iCm7WwTR

fcc919c84d2be6…217ea9e6d298f0

5 in → 5 out12.9183 XPM885 B

AeFp688r…Ec7aJVQr AWjJtsNe…scZ5ZBiH ANWcCA1W…FYxSHbqJ AXqtznv3…qHK3LfRr ANKZ8GNu…xk4HrFYE

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.

9.519 × 10⁹³(94-digit number)

95190320318736612503…70048625094465666559

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

9.519 × 10⁹³(94-digit number)

95190320318736612503…70048625094465666559

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

9.519 × 10⁹³(94-digit number)

95190320318736612503…70048625094465666561

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

1.903 × 10⁹⁴(95-digit number)

19038064063747322500…40097250188931333119

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.903 × 10⁹⁴(95-digit number)

19038064063747322500…40097250188931333121

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

3.807 × 10⁹⁴(95-digit number)

38076128127494645001…80194500377862666239

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.807 × 10⁹⁴(95-digit number)

38076128127494645001…80194500377862666241

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

7.615 × 10⁹⁴(95-digit number)

76152256254989290003…60389000755725332479

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.615 × 10⁹⁴(95-digit number)

76152256254989290003…60389000755725332481

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

1.523 × 10⁹⁵(96-digit number)

15230451250997858000…20778001511450664959

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.523 × 10⁹⁵(96-digit number)

15230451250997858000…20778001511450664961

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