Block #162,213

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 9/13/2013, 4:08:41 AM · Difficulty 9.8610 · 6,633,407 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

6c1dce96f00b9f348a82b4c459cf3294ac0e9a8add0f7b619b5afc44253bbdf2

View on Chainz ↗

Height

#162,213

Difficulty

9.860998

Transactions

Size

16.46 KB

Version

Bits

09dc6a58

Nonce

123,459

Timestamp

9/13/2013, 4:08:41 AM

Confirmations

6,633,407

Mined by

⛏️ P2SHAKvSRhswV6UTULx43VN4csQJXRNqqnvkd6

Merkle Root

98ee0118375d95cad14b586374309838a8e3dde1608f3a81627d8d3b75c9b73f

Transactions (5)

Coinbase1ff66e1acefa56…056df8af3a6f71

1 in → 1 out10.4700 XPM109 B

AKvSRhsw…Nqqnvkd6

4140f326b50db7…43b65ed5552e9b

140 in → 2 out1518.4600 XPM15.66 KB

AMwewRiE…afEnxySj ANLm86MU…cjLmdcRw

ebd26b2b28063d…5b3a656d12095e

1 in → 1 out10.2700 XPM159 B

AQt7xHtg…AXmNFrUW

bbe3629671458b…e49b18cab12aba

1 in → 1 out10.2600 XPM159 B

AQt7xHtg…AXmNFrUW

39ae28d1130a71…1bba8ccd710c46

2 in → 1 out10.4000 XPM306 B

AUV7kFuZ…v2E22BPs

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.

9.410 × 10⁹⁵(96-digit number)

94105725387745974241…56406970309666213919

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

9.410 × 10⁹⁵(96-digit number)

94105725387745974241…56406970309666213919

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

9.410 × 10⁹⁵(96-digit number)

94105725387745974241…56406970309666213921

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

1.882 × 10⁹⁶(97-digit number)

18821145077549194848…12813940619332427839

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.882 × 10⁹⁶(97-digit number)

18821145077549194848…12813940619332427841

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

3.764 × 10⁹⁶(97-digit number)

37642290155098389696…25627881238664855679

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.764 × 10⁹⁶(97-digit number)

37642290155098389696…25627881238664855681

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

7.528 × 10⁹⁶(97-digit number)

75284580310196779392…51255762477329711359

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.528 × 10⁹⁶(97-digit number)

75284580310196779392…51255762477329711361

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

1.505 × 10⁹⁷(98-digit number)

15056916062039355878…02511524954659422719

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.505 × 10⁹⁷(98-digit number)

15056916062039355878…02511524954659422721

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