Block #3,413,181

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.

1.744 × 10⁹⁴(95-digit number)

17444416594885967336…76547613402787017719

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

1.744 × 10⁹⁴(95-digit number)

17444416594885967336…76547613402787017719

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.744 × 10⁹⁴(95-digit number)

17444416594885967336…76547613402787017721

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

3.488 × 10⁹⁴(95-digit number)

34888833189771934673…53095226805574035439

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.488 × 10⁹⁴(95-digit number)

34888833189771934673…53095226805574035441

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

6.977 × 10⁹⁴(95-digit number)

69777666379543869347…06190453611148070879

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.977 × 10⁹⁴(95-digit number)

69777666379543869347…06190453611148070881

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

1.395 × 10⁹⁵(96-digit number)

13955533275908773869…12380907222296141759

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.395 × 10⁹⁵(96-digit number)

13955533275908773869…12380907222296141761

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

2.791 × 10⁹⁵(96-digit number)

27911066551817547738…24761814444592283519

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.791 × 10⁹⁵(96-digit number)

27911066551817547738…24761814444592283521

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 #3,413,181

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