Block #270,053

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/23/2013, 4:14:44 PM · Difficulty 9.9519 · 6,525,808 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

21d741ac70576e1bb165b4ba38eee8a32787de9260201e1be488ce4e0d9bd3a6

View on Chainz ↗

Height

#270,053

Difficulty

9.951854

Transactions

Size

426 B

Version

Bits

09f3acb3

Nonce

82,317

Timestamp

11/23/2013, 4:14:44 PM

Confirmations

6,525,808

Mined by

⛏️ P2SHASkLdRGQoNJxhyaLvsXPpng2V8ZUuwbaVe

Merkle Root

bf0a482ec35b028d704824c7ed1c52287455ed70334b094d6e0d0c504bfc6034

Transactions (2)

Coinbase0f7897aaa900d5…8325ad7c762142

1 in → 1 out10.0900 XPM109 B

ASkLdRGQ…ZUuwbaVe

08ae6768ff248a…5cfbba9f3b2014

1 in → 2 out6.0139 XPM226 B

AbSYvRZW…DrpTULpi ANHpjSPm…gwPN1e9g

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.

1.586 × 10⁹⁷(98-digit number)

15864944740689115693…59198751754752931639

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

1.586 × 10⁹⁷(98-digit number)

15864944740689115693…59198751754752931639

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.586 × 10⁹⁷(98-digit number)

15864944740689115693…59198751754752931641

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

3.172 × 10⁹⁷(98-digit number)

31729889481378231387…18397503509505863279

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.172 × 10⁹⁷(98-digit number)

31729889481378231387…18397503509505863281

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

6.345 × 10⁹⁷(98-digit number)

63459778962756462774…36795007019011726559

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.345 × 10⁹⁷(98-digit number)

63459778962756462774…36795007019011726561

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

1.269 × 10⁹⁸(99-digit number)

12691955792551292554…73590014038023453119

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.269 × 10⁹⁸(99-digit number)

12691955792551292554…73590014038023453121

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

2.538 × 10⁹⁸(99-digit number)

25383911585102585109…47180028076046906239

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