Block #164,924

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/14/2013, 11:14:45 PM · Difficulty 9.8645 · 6,632,890 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f42ea7d4835da5bb1f04bd0877e23702c985830622b08324029f6f1906bd6e85

View on Chainz ↗

Height

#164,924

Difficulty

9.864493

Transactions

Size

1.13 KB

Version

Bits

09dd4f66

Nonce

776,981

Timestamp

9/14/2013, 11:14:45 PM

Confirmations

6,632,890

Mined by

⛏️ P2SHAcPZnoF2QyAgP4PFHoxT7YzYksftaDynB5

Merkle Root

c84e623beb102904dd2ad6dbbb7defc3f99f075d1b96286d596f1d22c422e151

Transactions (2)

Coinbasef1db7d3c686a53…4c1a265fa0292a

1 in → 1 out10.2700 XPM109 B

AcPZnoF2…ftaDynB5

917a17c6676efe…037efec908c72c

6 in → 2 out2.0522 XPM964 B

AVg3BYuR…KTQNAxYY AFuBT9Xk…MEDBBhYN

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.

6.140 × 10⁹²(93-digit number)

61406151747989172214…21043199094597549999

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

6.140 × 10⁹²(93-digit number)

61406151747989172214…21043199094597549999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

6.140 × 10⁹²(93-digit number)

61406151747989172214…21043199094597550001

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

1.228 × 10⁹³(94-digit number)

12281230349597834442…42086398189195099999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.228 × 10⁹³(94-digit number)

12281230349597834442…42086398189195100001

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

2.456 × 10⁹³(94-digit number)

24562460699195668885…84172796378390199999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.456 × 10⁹³(94-digit number)

24562460699195668885…84172796378390200001

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

4.912 × 10⁹³(94-digit number)

49124921398391337771…68345592756780399999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.912 × 10⁹³(94-digit number)

49124921398391337771…68345592756780400001

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

9.824 × 10⁹³(94-digit number)

98249842796782675542…36691185513560799999

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