Gapcoin Cryptocurrency: A PeertoPeer Electronic Cash System
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PRIME GAP OF MAXIMUM KNOWN MERIT  PRESS RELEASE  MARCH 2018
Q. Now write the expression for the volume of a thick crust pizza with height "a" and radius "z". *

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What is Gapcoin?
Gapcoin is a prime number based p2p cryptocurrency, which eliminates the sticking points of other scientific currencies like Primecoin or Riecoin. It's a fork of Satoshi Nakamoto's Bitcoin, a decentralized payment system which is independent of banks, governments and other centralized regulators. With Gapcoin, you can send money around the globe in seconds. The big improvement in comparison to Bitcoin is that instead of burning electricity for it's own sake, Gapcoins Proof of Work function actually does useful work by searching for large prime gaps.
Specifications:
Fair launch:
Gapcoin was not designed to enrich the early adopters or the coin creators! Unlike Primecoin, the more people that mine Gapcoin, the more coins per block will be produced. (Coin supply will increase logarithmically with the difficulty, this means it will grow in the beginning, but later, it won't change that much.)
There wasn't any premine!
To avoid instamine, the reward of the first 1152 blocks (about 48 hours) increased quadratically to it's absolute value: the current difficulty. The block reward was 1/1152^2 * blockheight^2 * difficulty for the first 1152 blocks. The source code was made available before launch (excluding the PoW function), so that everyone could setup their own testing environment, compile the software, and check that everything works. Windows and Linux binaries were distributed in an encrypted container before launch, the password was revealed at launch on Tue Oct. 21 2014  18:00:00 UTC
How is Proof of Work actually designed?
A PoW algorithm has to fit two specifications:
Verifying a prime gap is easy, you only have to check every number between the start and the end to be composite. Calculating is harder, much harder! Large prime gaps occur a lot lesser than smaller ones. According to E. Westzynthius, in e^n prime gaps there will be one gap that is n times greater than the average prime gap.
So the difficulty will simply be the length of the prime gap?
Not exactly. The average length of a prime gap with the starting prime p, is log(p), which means that the average prime gap size increases with lager primes. Then, instead of the pure length, we use the merit of the prime gap, which is the ratio of the gap's size to the average gap size.
Let p be the prime starting a prime gap, then m = gapsize/log(p) will be the merit of this prime gap.
Also a pseudo random number is calculated from p to provide finer difficulty adjustment.
Let rand(p) be a pseudo random function with 0 less than rand(p) less than 1. Then, for a prime gap starting at prime p with size s, the difficulty will be s/log(p) + 2/log(p) * rand(p), where 2/log(p) is the average distance between a gap of size s and s + 2 (the next greater gap) in the proximity of p.
When it actually comes to mining, there are two additional fields added to the Blockheader, named “shift” and “adder”. We will calculate the prime p as sha256(Blockheader) * 2^shift + adder. As an additional criterion the adder has to be smaller than 2^shift to avoid that the PoW could be reused.
Could we break any world records?
In early 2018 the Gapcoin network found a new prime gap of maximum known merit! The merit M=G/ln(P1) of this gap is M=41.93878373153988, the largest merit of any known prime gap, and the first prime gap to be discovered with a merit exceeding 40. Source : http://www.trnicely.net/#MaxMerit
We already broke more than 544 records of first known occurrence prime gaps. (Note: New record data to be updated from the blockchain).
Also, if the difficulty reaches 41.94, every block will be a new world record!
But how are prime gaps useful?
Prime numbers are interesting for lots of mathematicians around the globe, and they're also important to every day cryptography (see RSA). Researches about prime gaps could not only lead to new breakthroughs in the bounded gap, it may also help proving the Twin Prime Conjecture and maybe even the millennium problem. Who knows?
Last but not least, what does the original Gapcoin logo have to do with prime gaps?
The formula π(x) is known as the primecounting function, the graph shows the prime gap distribution as well. That's why the “π” fits the logo perfectly. (image Wikipedia)
Source: 'Official' Gapcoin website: gapcoin.org
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* A. The formula for volume is π·(radius)2·(height). In this case, pi·z·z·a !
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