Block #82,865

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 7/25/2013, 5:47:36 PM · Difficulty 9.2730 · 6,706,917 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

3ec4adc5a6b6649524a278c855aafd2519749b15ce76c34892010881f7386024

View on Chainz ↗

Height

#82,865

Difficulty

9.272952

Transactions

Size

1.09 KB

Version

Bits

0945e02b

Nonce

17,000

Timestamp

7/25/2013, 5:47:36 PM

Confirmations

6,706,917

Merkle Root

cfa6535f0d6574b2cf702050b267f3abdafa0d0ae380366dde115a688adf74d3

Transactions (5)

Coinbaseedb1f51bbc6a0e…02afe66381be4b

1 in → 1 out11.6500 XPM109 B

AdZa3TAm…cTySUJQt

09bb8148e3dfa2…1a7956a320d70f

1 in → 2 out11.6800 XPM192 B

APmHtoxR…1yeEB9nC Aa3Fe5M5…B9tp95mD

fc5efe0fb28b69…8a4fd45178658c

1 in → 2 out11.6800 XPM191 B

Aa3Fe5M5…B9tp95mD AXaisQby…p4EH92L3

3fd97b292d74b6…8b0b594715a1f5

1 in → 1 out11.9000 XPM192 B

AX662CwP…jWnfrfSw

70db9a07ca704b…51af2640f6805a

2 in → 1 out3.3500 XPM340 B

Aa3Fe5M5…B9tp95mD

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.

5.385 × 10¹⁰²(103-digit number)

53859899323876350553…31277874469374852339

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

5.385 × 10¹⁰²(103-digit number)

53859899323876350553…31277874469374852339

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.385 × 10¹⁰²(103-digit number)

53859899323876350553…31277874469374852341

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

1.077 × 10¹⁰³(104-digit number)

10771979864775270110…62555748938749704679

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.077 × 10¹⁰³(104-digit number)

10771979864775270110…62555748938749704681

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

2.154 × 10¹⁰³(104-digit number)

21543959729550540221…25111497877499409359

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.154 × 10¹⁰³(104-digit number)

21543959729550540221…25111497877499409361

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

4.308 × 10¹⁰³(104-digit number)

43087919459101080442…50222995754998818719

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.308 × 10¹⁰³(104-digit number)

43087919459101080442…50222995754998818721

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

8.617 × 10¹⁰³(104-digit number)

86175838918202160885…00445991509997637439

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