Block #2,709,213

TWNLength 11★★★☆☆

Bi-Twin Chain · Discovered 6/17/2018, 4:57:35 PM · Difficulty 11.5918 · 4,134,126 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

faf7867f4b72cbb43b96fa4aa0252d70525aa563ceab104864659a590bb97515

View on Chainz ↗

Height

#2,709,213

Difficulty

11.591787

Transactions

Size

575 B

Version

Bits

0b977f58

Nonce

955,696,367

Timestamp

6/17/2018, 4:57:35 PM

Confirmations

4,134,126

Mined by

⛏️ P2SHAJkEtPc9GgJSpgsWUxSqFcF1DhZByutE24

Merkle Root

cee2bf9da34935bf418207b17406a8a232ecedbd3c9a37b36f197d7282387d09

Transactions (2)

Coinbase853965c91bd4a1…5d6de8f8a52035

1 in → 1 out7.4400 XPM110 B

AJkEtPc9…ZByutE24

49247c1bc7ebd0…a29d4636392ef5

2 in → 2 out6.3020 XPM373 B

AGmaxG8p…8bPXJ8m7 AHM4Lkw2…NaayHpzT

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.

7.553 × 10⁹⁹(100-digit number)

75537062619883509472…26626789501380198399

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

7.553 × 10⁹⁹(100-digit number)

75537062619883509472…26626789501380198399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.553 × 10⁹⁹(100-digit number)

75537062619883509472…26626789501380198401

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

1.510 × 10¹⁰⁰(101-digit number)

15107412523976701894…53253579002760396799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.510 × 10¹⁰⁰(101-digit number)

15107412523976701894…53253579002760396801

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

3.021 × 10¹⁰⁰(101-digit number)

30214825047953403789…06507158005520793599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.021 × 10¹⁰⁰(101-digit number)

30214825047953403789…06507158005520793601

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

6.042 × 10¹⁰⁰(101-digit number)

60429650095906807578…13014316011041587199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.042 × 10¹⁰⁰(101-digit number)

60429650095906807578…13014316011041587201

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

1.208 × 10¹⁰¹(102-digit number)

12085930019181361515…26028632022083174399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.208 × 10¹⁰¹(102-digit number)

12085930019181361515…26028632022083174401

Verify on FactorDB ↗Wolfram Alpha ↗

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

Level 5 — Twin Prime Pair (2^5 × origin ± 1)

2^5 × origin − 1

2.417 × 10¹⁰¹(102-digit number)

24171860038362723031…52057264044166348799

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 11 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

RareChain length 11

Approximately 1 in 1,000 blocks. Noteworthy discoveries.

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 #2,709,213

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