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Block #157,939

TWNLength 9β˜…β˜†β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 9/9/2013, 11:50:06 PM Β· Difficulty 9.8689 Β· 6,656,541 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
c457f8d34c0e368e3b9d1284c708f93d8a3dd93190170bf3f3243206120ed62c
View on Chainz β†—

Height

#157,939

Difficulty

9.868911

Transactions

1

Size

200 B

Version

2

Bits

09de70f8

Nonce

66,414

Timestamp

9/9/2013, 11:50:06 PM

Confirmations

6,656,541

Mined by

⛏️ P2SHAQE5juVGf2GjfXcDEExb1YTp8cv2LcRcDH

Merkle Root

70822982d5377bba1b0dc94c1bd54ca2f1dbac6f6b1ee92acafabf64a0c4c24d
Transactions (1)
Coinbase70822982d5377b…fabf64a0c4c24d
1 in β†’ 1 out10.2500 XPM109 B
AQE5juVG…v2LcRcDH
Prime Chain Origin

This is the prime chain origin stored in the block header. It is a composite number (not prime itself) β€” it equals the first prime in the chain multiplied by a primorial. The origin anchors the entire chain to this specific block.

3.794 Γ— 10⁹⁢(97-digit number)
37949492117231714418…02052592921372262399
Discovered Prime Numbers
Lower: 2^k Γ— origin βˆ’ 1 | Upper: 2^k Γ— 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 β€” Twin Prime Pair (origin Β± 1)
origin βˆ’ 1
3.794 Γ— 10⁹⁢(97-digit number)
37949492117231714418…02052592921372262399
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
3.794 Γ— 10⁹⁢(97-digit number)
37949492117231714418…02052592921372262401
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: origin + 1 βˆ’ origin βˆ’ 1 = 2 (twin primes βœ“)
Level 1 β€” Twin Prime Pair (2^1 Γ— origin Β± 1)
2^1 Γ— origin βˆ’ 1
7.589 Γ— 10⁹⁢(97-digit number)
75898984234463428836…04105185842744524799
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
7.589 Γ— 10⁹⁢(97-digit number)
75898984234463428836…04105185842744524801
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^1 Γ— origin + 1 βˆ’ 2^1 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 2 β€” Twin Prime Pair (2^2 Γ— origin Β± 1)
2^2 Γ— origin βˆ’ 1
1.517 Γ— 10⁹⁷(98-digit number)
15179796846892685767…08210371685489049599
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
1.517 Γ— 10⁹⁷(98-digit number)
15179796846892685767…08210371685489049601
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^2 Γ— origin + 1 βˆ’ 2^2 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 3 β€” Twin Prime Pair (2^3 Γ— origin Β± 1)
2^3 Γ— origin βˆ’ 1
3.035 Γ— 10⁹⁷(98-digit number)
30359593693785371534…16420743370978099199
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
3.035 Γ— 10⁹⁷(98-digit number)
30359593693785371534…16420743370978099201
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^3 Γ— origin + 1 βˆ’ 2^3 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)
Level 4 β€” Twin Prime Pair (2^4 Γ— origin Β± 1)
2^4 Γ— origin βˆ’ 1
6.071 Γ— 10⁹⁷(98-digit number)
60719187387570743068…32841486741956198399
Verify on FactorDB β†—Wolfram Alpha β†—

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 β€” the large number shown above β€” anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

β˜…β˜†β˜†β˜†β˜†
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 Γ— 3 Γ— 5 Γ— 7 Γ— …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial β€” that divisibility is part of the proof.

Prime Chain Origin = First Prime Γ— Primorial (2Β·3Β·5Β·7Β·11·…)
Source: Primecoin prime.cpp β€” CheckPrimeProofOfWork()

This is why the origin has many small prime factors β€” 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 βˆ’ 1 and p+2 = origin/primorial + 1
Cross-reference on Chainz Explorer
Circulating Supply:57,759,915 XPMΒ·at block #6,814,479 Β· updates every 60s
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