Block #81,839

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 7/25/2013, 12:58:28 AM · Difficulty 9.2704 · 6,712,789 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

f588902da54d864115ed47ee26ff15d405a4ff057a8a34917e25348cac86240b

View on Chainz ↗

Height

#81,839

Difficulty

9.270398

Transactions

Size

765 B

Version

Bits

094538c6

Nonce

37,129

Timestamp

7/25/2013, 12:58:28 AM

Confirmations

6,712,789

Mined by

⛏️ P2SHAd3mbBEA5seWCD6zwvCAa1UUjeSVGhuUmg

Merkle Root

586eca31d6f2d45651640e791a8fea5652348e102488ad1bd9c205d5d98f3484

Transactions (3)

Coinbase2a6f5698f1fef7…23dd2466777e20

1 in → 1 out11.6400 XPM109 B

Ad3mbBEA…SVGhuUmg

94bb405cbd2dfd…de2ee4106c3070

2 in → 2 out324.3200 XPM374 B

Aad9kqsq…hcSyhTzF AV7LDZ9h…LucHuVTS

bc561c13eee36f…8ffb4b1d79c1f0

1 in → 1 out239.9900 XPM191 B

AP6mrFkf…AbQY7Uw9

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.

8.342 × 10⁹⁷(98-digit number)

83426673768253971822…44369807836764850499

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

8.342 × 10⁹⁷(98-digit number)

83426673768253971822…44369807836764850499

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.342 × 10⁹⁷(98-digit number)

83426673768253971822…44369807836764850501

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

1.668 × 10⁹⁸(99-digit number)

16685334753650794364…88739615673529700999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.668 × 10⁹⁸(99-digit number)

16685334753650794364…88739615673529701001

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

3.337 × 10⁹⁸(99-digit number)

33370669507301588729…77479231347059401999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.337 × 10⁹⁸(99-digit number)

33370669507301588729…77479231347059402001

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

6.674 × 10⁹⁸(99-digit number)

66741339014603177458…54958462694118803999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.674 × 10⁹⁸(99-digit number)

66741339014603177458…54958462694118804001

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

1.334 × 10⁹⁹(100-digit number)

13348267802920635491…09916925388237607999

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