Block #410,052

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/18/2014, 7:12:04 PM · Difficulty 10.4299 · 6,395,636 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f9441c855349dde48c73ad9de981d83cd7a6c994ae76bd47e5c826b97094022c

View on Chainz ↗

Height

#410,052

Difficulty

10.429911

Transactions

Size

1.25 KB

Version

Bits

0a6e0ea4

Nonce

103,229

Timestamp

2/18/2014, 7:12:04 PM

Confirmations

6,395,636

Mined by

⛏️ AaSAAQvZBsUoAqCaKecR1VEueWTuBchppgRZTJ

Merkle Root

429e6010ff4675a83088911fc6bf7150aa4e3602975f3b8bac323ca53a615bd0

Transactions (2)

Coinbasece55d5be62ab26…0b4be3ded0e5e5

1 in → 22 out9.1800 XPM811 B

AQvZBsUo…hppgRZTJ Aac9Hjdg…P4ktypc3 AGKhfQo2…h7fBXLX6 ALqxX1WA…hsKjbnXn AYtDvcGs…VuAcA7m7

c877df8249787a…cc6f5b37b71707

2 in → 2 out1.2941 XPM374 B

AbwGkGTq…KrSkJ65d Ab45HuDa…PBaTv6sz

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.

5.411 × 10⁹⁸(99-digit number)

54114060368700164528…09958475818537830399

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

5.411 × 10⁹⁸(99-digit number)

54114060368700164528…09958475818537830399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.411 × 10⁹⁸(99-digit number)

54114060368700164528…09958475818537830401

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

1.082 × 10⁹⁹(100-digit number)

10822812073740032905…19916951637075660799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.082 × 10⁹⁹(100-digit number)

10822812073740032905…19916951637075660801

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

2.164 × 10⁹⁹(100-digit number)

21645624147480065811…39833903274151321599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.164 × 10⁹⁹(100-digit number)

21645624147480065811…39833903274151321601

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

4.329 × 10⁹⁹(100-digit number)

43291248294960131623…79667806548302643199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.329 × 10⁹⁹(100-digit number)

43291248294960131623…79667806548302643201

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

8.658 × 10⁹⁹(100-digit number)

86582496589920263246…59335613096605286399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

8.658 × 10⁹⁹(100-digit number)

86582496589920263246…59335613096605286401

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