Block #165,830

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.

7.252 × 10⁹⁸(99-digit number)

72521547463496514794…22760001139282752849

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

7.252 × 10⁹⁸(99-digit number)

72521547463496514794…22760001139282752849

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.252 × 10⁹⁸(99-digit number)

72521547463496514794…22760001139282752851

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

1.450 × 10⁹⁹(100-digit number)

14504309492699302958…45520002278565505699

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.450 × 10⁹⁹(100-digit number)

14504309492699302958…45520002278565505701

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

2.900 × 10⁹⁹(100-digit number)

29008618985398605917…91040004557131011399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.900 × 10⁹⁹(100-digit number)

29008618985398605917…91040004557131011401

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

5.801 × 10⁹⁹(100-digit number)

58017237970797211835…82080009114262022799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.801 × 10⁹⁹(100-digit number)

58017237970797211835…82080009114262022801

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

1.160 × 10¹⁰⁰(101-digit number)

11603447594159442367…64160018228524045599

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