Block #213,461

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 10/16/2013, 9:27:36 PM · Difficulty 9.9216 · 6,594,846 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

722ddad193b2541a003cc16d880e2e8388e1279962866b7440c43f10b0987860

View on Chainz ↗

Height

#213,461

Difficulty

9.921628

Transactions

Size

1.01 KB

Version

Bits

09ebefd4

Nonce

21,943

Timestamp

10/16/2013, 9:27:36 PM

Confirmations

6,594,846

Mined by

⛏️ P2SHAbuU1HhSi58QcdzhwKqhpJiRG3JPwsJEbB

Merkle Root

3bb83971e36afe72f3e4872be643e90b808a836e2ce8dbdf92977e93ed002a00

Transactions (5)

Coinbase123525528652d3…caafcaac7957d6

1 in → 1 out10.1800 XPM110 B

AbuU1HhS…JPwsJEbB

456790bdfb6025…597ae699338f06

1 in → 1 out10.1700 XPM159 B

ANDgnMfg…8bhqRjDp

856d5862727bce…b87ffff769f618

1 in → 2 out10.1500 XPM226 B

AXxYaivX…wvXsm3LH AaVFHFSq…bCnX1Bpw

4dda5a7f672fd3…c2f7644276a7c4

1 in → 2 out10.1500 XPM227 B

AR5tgCUG…U4284hTS Ab3dSvM7…Qoxxb9tp

e31f7bae4e3b84…de265c51f16d68

1 in → 2 out7.1255 XPM226 B

APSAumdD…KmgnVafy AdrWK3RZ…g3imBxiN

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.

2.183 × 10⁹³(94-digit number)

21836377156976924316…17562869434653476799

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

2.183 × 10⁹³(94-digit number)

21836377156976924316…17562869434653476799

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.183 × 10⁹³(94-digit number)

21836377156976924316…17562869434653476801

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

4.367 × 10⁹³(94-digit number)

43672754313953848633…35125738869306953599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.367 × 10⁹³(94-digit number)

43672754313953848633…35125738869306953601

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

8.734 × 10⁹³(94-digit number)

87345508627907697266…70251477738613907199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.734 × 10⁹³(94-digit number)

87345508627907697266…70251477738613907201

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

1.746 × 10⁹⁴(95-digit number)

17469101725581539453…40502955477227814399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.746 × 10⁹⁴(95-digit number)

17469101725581539453…40502955477227814401

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

3.493 × 10⁹⁴(95-digit number)

34938203451163078906…81005910954455628799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.493 × 10⁹⁴(95-digit number)

34938203451163078906…81005910954455628801

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

Block #213,461

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