Block #261,856

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/16/2013, 6:03:21 AM · Difficulty 9.9704 · 6,554,453 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

230135180d63109346a1780c6662f2c0e6a22864d672d6a95459e3a521f8dd7c

View on Chainz ↗

Height

#261,856

Difficulty

9.970420

Transactions

Size

1.94 KB

Version

Bits

09f86d70

Nonce

93,315

Timestamp

11/16/2013, 6:03:21 AM

Confirmations

6,554,453

Mined by

⛏️ aRAJ83QDaoSnQQgeKxvuLGszzGf7DqWdAVWo

Merkle Root

e30bb070f216683123e41e31df4d4bef71efbe1421c6640bd7c7796fa926b010

Transactions (1)

Coinbasee30bb070f21668…c7796fa926b010

1 in → 54 out10.0200 XPM1.85 KB

AJ83QDao…DqWdAVWo AFqKfjez…qX9Ace3g Ab2f51dk…irxE26FW APH8rV6F…heb1HF2Z AQtzUSwq…htAuF3yC

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.

2.130 × 10⁹³(94-digit number)

21307378093249775839…58397842216786967999

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

2.130 × 10⁹³(94-digit number)

21307378093249775839…58397842216786967999

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.130 × 10⁹³(94-digit number)

21307378093249775839…58397842216786968001

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

4.261 × 10⁹³(94-digit number)

42614756186499551679…16795684433573935999

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.261 × 10⁹³(94-digit number)

42614756186499551679…16795684433573936001

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

8.522 × 10⁹³(94-digit number)

85229512372999103359…33591368867147871999

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.522 × 10⁹³(94-digit number)

85229512372999103359…33591368867147872001

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

1.704 × 10⁹⁴(95-digit number)

17045902474599820671…67182737734295743999

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.704 × 10⁹⁴(95-digit number)

17045902474599820671…67182737734295744001

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

3.409 × 10⁹⁴(95-digit number)

34091804949199641343…34365475468591487999

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

Block #261,856

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