Block #413,128

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/21/2014, 12:40:31 AM · Difficulty 10.4160 · 6,402,927 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f2e8ddb4b19860612f5d6d59f146ee217a5c693aa9466d6dae926d0962ba80ad

View on Chainz ↗

Height

#413,128

Difficulty

10.416033

Transactions

Size

1.08 KB

Version

Bits

0a6a8123

Nonce

21,670

Timestamp

2/21/2014, 12:40:31 AM

Confirmations

6,402,927

Mined by

⛏️ P2SHAbwWkWZtvvKWFxiUbZcGD749fwkawDhufa

Merkle Root

be6346e002c648cf42bc79344988f24508f7f0493e7f76c1507c9fc7a041b5ef

Transactions (5)

Coinbase2c1feff94b79ee…266f12eef532eb

1 in → 1 out9.2400 XPM116 B

AbwWkWZt…kawDhufa

76b255b1a9524f…49476d8daa1374

1 in → 2 out6.1278 XPM226 B

ALdjDMf1…79i2CJci AJEQeGMP…k6k8RPUu

27f714b70347c9…c77c731c3b6987

1 in → 2 out5.0973 XPM226 B

ATvJbqH4…TwBJmvSu AGz5ycfC…gmvHMXuj

b0d7bdf4f6c99d…7c6f81d733a07a

1 in → 2 out1.7910 XPM226 B

AXar9Uwd…edwhxSZV AMU8nEj9…Eo32pqxr

222d734ef9d5a3…9f97c00c69573d

1 in → 2 out3.0662 XPM227 B

AXxp24rY…yi3yUFBu AVzppJKF…3pBdsio3

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.

4.465 × 10⁹¹(92-digit number)

44651596464654730414…89337388081850243199

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

4.465 × 10⁹¹(92-digit number)

44651596464654730414…89337388081850243199

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.465 × 10⁹¹(92-digit number)

44651596464654730414…89337388081850243201

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

8.930 × 10⁹¹(92-digit number)

89303192929309460829…78674776163700486399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.930 × 10⁹¹(92-digit number)

89303192929309460829…78674776163700486401

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

1.786 × 10⁹²(93-digit number)

17860638585861892165…57349552327400972799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.786 × 10⁹²(93-digit number)

17860638585861892165…57349552327400972801

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

3.572 × 10⁹²(93-digit number)

35721277171723784331…14699104654801945599

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.572 × 10⁹²(93-digit number)

35721277171723784331…14699104654801945601

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

7.144 × 10⁹²(93-digit number)

71442554343447568663…29398209309603891199

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

7.144 × 10⁹²(93-digit number)

71442554343447568663…29398209309603891201

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

Block #413,128

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