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Block #31,378

TWNLength 8β˜…β˜†β˜†β˜†β˜†

Bi-Twin Chain Β· Discovered 7/13/2013, 11:12:02 PM Β· Difficulty 7.9886 Β· 6,775,512 confirmations

TWN
Bi-Twin Chain

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

Block Header
Block Hash
211b865765b14dff2fcbb16a1833a2f4c9023510534e7c38dbc98e249fd25b75
View on Chainz β†—

Height

#31,378

Difficulty

7.988587

Transactions

1

Size

200 B

Version

2

Bits

07fd1411

Nonce

197

Timestamp

7/13/2013, 11:12:02 PM

Confirmations

6,775,512

Mined by

⛏️ P2SHAV6P7enUMtWL6sjxBf1aCsLJbsVLHeixvR

Merkle Root

32d3dfd22bf9666b27a2b386d0e7df21ef7b12eecea95ccd9ae3ac62abe61e80
Transactions (1)
Coinbase32d3dfd22bf966…e3ac62abe61e80
1 in β†’ 1 out15.6500 XPM108 B
AV6P7enU…VLHeixvR
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.

9.745 Γ— 10⁹⁹(100-digit number)
97455596001324525916…03752573209406958399
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
9.745 Γ— 10⁹⁹(100-digit number)
97455596001324525916…03752573209406958399
Verify on FactorDB β†—Wolfram Alpha β†—
origin + 1
9.745 Γ— 10⁹⁹(100-digit number)
97455596001324525916…03752573209406958401
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
1.949 Γ— 10¹⁰⁰(101-digit number)
19491119200264905183…07505146418813916799
Verify on FactorDB β†—Wolfram Alpha β†—
2^1 Γ— origin + 1
1.949 Γ— 10¹⁰⁰(101-digit number)
19491119200264905183…07505146418813916801
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
3.898 Γ— 10¹⁰⁰(101-digit number)
38982238400529810366…15010292837627833599
Verify on FactorDB β†—Wolfram Alpha β†—
2^2 Γ— origin + 1
3.898 Γ— 10¹⁰⁰(101-digit number)
38982238400529810366…15010292837627833601
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
7.796 Γ— 10¹⁰⁰(101-digit number)
77964476801059620732…30020585675255667199
Verify on FactorDB β†—Wolfram Alpha β†—
2^3 Γ— origin + 1
7.796 Γ— 10¹⁰⁰(101-digit number)
77964476801059620732…30020585675255667201
Verify on FactorDB β†—Wolfram Alpha β†—
Difference: 2^3 Γ— origin + 1 βˆ’ 2^3 Γ— origin βˆ’ 1 = 2 (twin primes βœ“)

What this block proved

The miner who found this block proved the existence of 8 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 8

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:57,699,228 XPMΒ·at block #6,806,889 Β· updates every 60s
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AQZeJoQ81VgeHQBpEYTgvNqV1gnz3L5SvK

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