Earlier this month, we brought you the news that **University of New Hampshire mathematician Yitang Zhang proved a weak version of the twin prime conjecture**, showing that there is an endless supply of prime pairs which are separated by no more than 70,000,000. That seems like it’s a long way from proving the proper twin prime conjecture — showing an infinite number of primes separated by just two digits — but it might not be. **In their latest video, the math whizzes at Numberphile offer a great layman’s explanation of what the twin prime conjecture is and what Zhang’s work means for it. **

It turns out, when you’re dealing with the huge numbers that mathematicians work with, 70,000,000 and 2 aren’t really as far apart as they might seem. Huge numbers like googolplex can really mess with sense of scale that way. The Numberphile crew even points out that some researchers familiar with Zhang’s work see it as a stepping stone that could used to prove an infinite number of primes with as few as 16 numbers between them without a lot of reworking.

(via Numberphile)

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The **twin prime conjecture** states that there are an infinite number of twin primes — prime numbers where one prime is two greater than the other. For example, 1 and 3 are twin primes, so are 101 and 103, 881 and 883, and so on. **UPDATE: **As pointed out in the comments, 1 is not a prime! You can learn more about that little fact right here. This conjecture has never been proven, but **Yitang Zhang believes he has proven there are infinite primes paired with other primes no greater than 70 million more than themselves**. Here’s why that’s kind of a big deal.

Zhang hasn’t proven an infinite number of true twin primes where one prime is two greater than the other, but he’s proven that there are paired primes separated by other values up to 70 million. Instead of proving the twin prime conjecture stating there are an infinite number of primes where p and p+2 are both prime, he has proven an infinite number of p and p + less than 70 million.

70 million and 2 are very different numbers, but Zhang’s work shows that there’s a limit to the distance between an infinite number of prime numbers, and that’s never been proven before. Also, the 70 million limit seems huge, but considering the new largest-known prime number is 17 million digits long, it’s really not that big in terms of the numbers mathematicians are dealing with.

Mathematicians have known for centuries that there are an infinite number of prime numbers, but the jury is still out on twin prime conjecture. Prime numbers tend to get further apart from each other the higher you go. There’s not a set pattern for it, but if you chart out known primes, you get something that looks almost like a loose spiral.

Zhang started working on the proof last summer when he had the idea. Since making it public, other mathematicians have been checking out Zhang’s work, and so far it holds up. The proof is currently under review, and if it passes muster it will be published in *The Annals of Mathematics*.

(via Phys Org, image via renoir_girl)

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The previous largest-known prime was 2^{43,112,609} – 1, which when written out is a little under 13 million digits long, but 2^{57,885,161} – 1 just crushed it, coming in at 17 million digits long. 2^{57,885,161} was discovered by **Curtis Cooper** at the **University of Central Missouri** and his discovery was part of the **Great Internet Mersenne Prime Search (GIMPS)**.

Mersenne was a monk, who in 1644 came up with what he believed to be a formula that would find prime numbers. He said 2^{p}-1 with *p* being itself a prime number would result in a larger prime. It doesn’t work with every number, but it gives a starting point for finding large primes. In fact, all 10 of the largest-known prime numbers are Mersenne primes discovered by GIMPS. Although it isn’t perfect, there’s clearly something to Mersenne’s formula.

The GIMPS project is run by volunteers who use computers to test Mersenne prime candidates, and it rewards its volunteers who discover new primes. Cooper is going to receive a prize of $3,000 for his work, but that’s nothing compared to the $150,000 prize being offered by the Electronic Frontier Foundation for the discovery of 100 million digit prime. They also have a $250,000 prize for a prime of one billion digits.

(via New Scientist, image via Koen Vereeken)

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