Block #264,599

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/18/2013, 8:26:50 PM · Difficulty 9.9640 · 6,530,276 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

39a8de7e0e36648ed6a436cb471b5729435cd2a4f3496b306c81057dace422bb

View on Chainz ↗

Height

#264,599

Difficulty

9.964033

Transactions

Size

2.55 KB

Version

Bits

09f6cadf

Nonce

61,089

Timestamp

11/18/2013, 8:26:50 PM

Confirmations

6,530,276

Mined by

⛏️ P2SHAKcbk49N7DP3nse7MJr3Ed6rZiGJbKa7j1

Merkle Root

bc1948e7b0ac483b694bca477e2db3037f829b1cd16eaf274057483e0deb7cf6

Transactions (5)

Coinbasec26ef3fd99fecb…fd689d10e2754a

1 in → 2 out10.1100 XPM137 B

AKcbk49N…GJbKa7j1 AKNbfPoc…jfkpwAyL

e03484f8a5ed2f…202adfa68e6420

2 in → 4 out10.3859 XPM409 B

AFsMW6dF…Z3wb5bBq AbNgL8yS…Wu6nPFk5 AeKNrjNj…1NEq95Ta AMuPGPXT…eMMNvd8v

6cb27c9a055fde…90cb0322486c51

1 in → 2 out3.2798 XPM227 B

AYhXSpJ5…MnkY7cdE ANUPa5nC…gS5uF4Pm

33a20d3db62e4b…6857623d77c7b9

2 in → 2 out1.1600 XPM374 B

AFofZMqs…tx9efDBr AahGmE2B…CG3ryD7f

bb97907e11f277…0496afca0e8a06

9 in → 1 out14.0500 XPM1.35 KB

AT6w2QXi…JFuHx8Sv

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.

4.391 × 10⁹⁶(97-digit number)

43913206312631379339…85685522065476554799

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

4.391 × 10⁹⁶(97-digit number)

43913206312631379339…85685522065476554799

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.391 × 10⁹⁶(97-digit number)

43913206312631379339…85685522065476554801

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

8.782 × 10⁹⁶(97-digit number)

87826412625262758678…71371044130953109599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.782 × 10⁹⁶(97-digit number)

87826412625262758678…71371044130953109601

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

1.756 × 10⁹⁷(98-digit number)

17565282525052551735…42742088261906219199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.756 × 10⁹⁷(98-digit number)

17565282525052551735…42742088261906219201

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

3.513 × 10⁹⁷(98-digit number)

35130565050105103471…85484176523812438399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.513 × 10⁹⁷(98-digit number)

35130565050105103471…85484176523812438401

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

7.026 × 10⁹⁷(98-digit number)

70261130100210206942…70968353047624876799

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