Block #829,451

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.

5.216 × 10⁹⁶(97-digit number)

52167247689083392308…13169253932452823039

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

5.216 × 10⁹⁶(97-digit number)

52167247689083392308…13169253932452823039

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.216 × 10⁹⁶(97-digit number)

52167247689083392308…13169253932452823041

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

1.043 × 10⁹⁷(98-digit number)

10433449537816678461…26338507864905646079

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.043 × 10⁹⁷(98-digit number)

10433449537816678461…26338507864905646081

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

2.086 × 10⁹⁷(98-digit number)

20866899075633356923…52677015729811292159

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.086 × 10⁹⁷(98-digit number)

20866899075633356923…52677015729811292161

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

4.173 × 10⁹⁷(98-digit number)

41733798151266713846…05354031459622584319

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.173 × 10⁹⁷(98-digit number)

41733798151266713846…05354031459622584321

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

8.346 × 10⁹⁷(98-digit number)

83467596302533427693…10708062919245168639

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

8.346 × 10⁹⁷(98-digit number)

83467596302533427693…10708062919245168641

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

1.669 × 10⁹⁸(99-digit number)

16693519260506685538…21416125838490337279

Verify on FactorDB ↗Wolfram Alpha ↗

2^5 × origin + 1

1.669 × 10⁹⁸(99-digit number)

16693519260506685538…21416125838490337281

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^5 × origin + 1 − 2^5 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 12 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

ExceptionalChain length 12

Around 1 in 10,000 blocks. A significant mathematical achievement.

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 #829,451

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