Block #413,033

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/20/2014, 10:50:04 PM · Difficulty 10.4173 · 6,383,803 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

d089962c47529362d558215283461fbeb5d458a297b53589032cc140d972c0b1

View on Chainz ↗

Height

#413,033

Difficulty

10.417328

Transactions

Size

968 B

Version

Bits

0a6ad5fd

Nonce

5,426

Timestamp

2/20/2014, 10:50:04 PM

Confirmations

6,383,803

Mined by

⛏️ iMaSAQvZBsUoAqCaKecR1VEueWTuBchppgRZTJ

Merkle Root

adca4d8860267ddff43cd768c40c8fbc3429b4f402d0f7b13741b719a39319c1

Transactions (1)

Coinbaseadca4d8860267d…41b719a39319c1

1 in → 24 out9.2000 XPM878 B

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

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.

4.200 × 10⁹⁵(96-digit number)

42008011645752326468…35936689923662069759

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

4.200 × 10⁹⁵(96-digit number)

42008011645752326468…35936689923662069759

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.200 × 10⁹⁵(96-digit number)

42008011645752326468…35936689923662069761

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

8.401 × 10⁹⁵(96-digit number)

84016023291504652937…71873379847324139519

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.401 × 10⁹⁵(96-digit number)

84016023291504652937…71873379847324139521

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

1.680 × 10⁹⁶(97-digit number)

16803204658300930587…43746759694648279039

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.680 × 10⁹⁶(97-digit number)

16803204658300930587…43746759694648279041

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

3.360 × 10⁹⁶(97-digit number)

33606409316601861174…87493519389296558079

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.360 × 10⁹⁶(97-digit number)

33606409316601861174…87493519389296558081

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

6.721 × 10⁹⁶(97-digit number)

67212818633203722349…74987038778593116159

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

6.721 × 10⁹⁶(97-digit number)

67212818633203722349…74987038778593116161

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