Block #851,765

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.

7.144 × 10⁹⁷(98-digit number)

71449851557826961163…39490937639890943999

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

7.144 × 10⁹⁷(98-digit number)

71449851557826961163…39490937639890943999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.144 × 10⁹⁷(98-digit number)

71449851557826961163…39490937639890944001

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

1.428 × 10⁹⁸(99-digit number)

14289970311565392232…78981875279781887999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.428 × 10⁹⁸(99-digit number)

14289970311565392232…78981875279781888001

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

2.857 × 10⁹⁸(99-digit number)

28579940623130784465…57963750559563775999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.857 × 10⁹⁸(99-digit number)

28579940623130784465…57963750559563776001

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

5.715 × 10⁹⁸(99-digit number)

57159881246261568930…15927501119127551999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.715 × 10⁹⁸(99-digit number)

57159881246261568930…15927501119127552001

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

1.143 × 10⁹⁹(100-digit number)

11431976249252313786…31855002238255103999

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.143 × 10⁹⁹(100-digit number)

11431976249252313786…31855002238255104001

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

Level 5 — Twin Prime Pair (2^5 × origin ± 1)

2^5 × origin − 1

2.286 × 10⁹⁹(100-digit number)

22863952498504627572…63710004476510207999

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 11 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

RareChain length 11

Approximately 1 in 1,000 blocks. Noteworthy discoveries.

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 #851,765

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