Block #266,412

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 11/20/2013, 10:25:45 AM · Difficulty 9.9606 · 6,533,118 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

29e4500224a9ec8989a29dbe8b01fcaaa7d539b041cd151e9e862ba00986e0a9

View on Chainz ↗

Height

#266,412

Difficulty

9.960621

Transactions

Size

1.28 KB

Version

Bits

09f5eb3f

Nonce

4,325

Timestamp

11/20/2013, 10:25:45 AM

Confirmations

6,533,118

Mined by

⛏️ P2SHAKcbk49N7DP3nse7MJr3Ed6rZiGJbKa7j1

Merkle Root

09d7f7a375ae823ae26c4b6deb38835e5d06904222b37833a8117bfec40adc96

Transactions (4)

Coinbasece5db6c8016903…7758b75daf77bf

1 in → 2 out10.0900 XPM141 B

AKcbk49N…GJbKa7j1 AKNbfPoc…jfkpwAyL

643728bb64d689…998c1f74be43cd

1 in → 2 out10.0500 XPM192 B

AHvZZByH…xAhtRkcq AP4WwkDc…Tr7pRNno

c476efa25a7e1d…8f1326b43513d4

3 in → 9 out30.1500 XPM659 B

AX2gzk4S…d2GsT5fq ASEVuzfF…Kzim7qGy AZCLzeSF…61wrGgPL AeKNrjNj…1NEq95Ta AWBkqPMM…QnvK8aGb

7a8d7d38dbe4d6…035cfb7052436e

1 in → 2 out5.0444 XPM226 B

AJD2MBFp…yQA75iRs AekfpJLM…D6M6cEs3

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.

2.510 × 10¹⁰²(103-digit number)

25102189905708121194…07824713385146872479

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

2.510 × 10¹⁰²(103-digit number)

25102189905708121194…07824713385146872479

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.510 × 10¹⁰²(103-digit number)

25102189905708121194…07824713385146872481

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

5.020 × 10¹⁰²(103-digit number)

50204379811416242388…15649426770293744959

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.020 × 10¹⁰²(103-digit number)

50204379811416242388…15649426770293744961

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

1.004 × 10¹⁰³(104-digit number)

10040875962283248477…31298853540587489919

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.004 × 10¹⁰³(104-digit number)

10040875962283248477…31298853540587489921

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

2.008 × 10¹⁰³(104-digit number)

20081751924566496955…62597707081174979839

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.008 × 10¹⁰³(104-digit number)

20081751924566496955…62597707081174979841

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

4.016 × 10¹⁰³(104-digit number)

40163503849132993911…25195414162349959679

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.016 × 10¹⁰³(104-digit number)

40163503849132993911…25195414162349959681

Verify on FactorDB ↗Wolfram Alpha ↗

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

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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