New Largest Known Prime Number

https://www.npr.org/2018/12/21/679207604/the-world-has-a-new-largest-known-prime-number

Life hack: Every prime number can be written as either 6n+1 or 6n-1.

Quote: RS

Life hack: Every prime number can be written as either 6n+1 or 6n-1.

Also, more details on the new largest prime
I was one of the people doing this a few years ago, but keep in mind that this ties up one of the cores of your computer for over a month (and it has to run 24/7), usually with the result, "No, it's not a prime."

For those of you who don't know how they find these, the Lucas-Lehmer test works on what are called Mersenne primes (primes of the form 2^P - 1, where P is a prime number - note that if P > 1 and not prime, 2^P - 1 cannot be prime).
Let X₀ = 4, and X_N+1 = (X_N² - 2) modulo (2^P - 1)
If X_(P-2) = 0, then 2^P - 1 is prime; otherwise, it isn't.

For example, for P = 5 (the number 31), the Xs are 4, (16 - 2) mod 31 = 14, (196 - 2) mod 31 = 8, and (64 - 2) mod 31 = 0; X₃ = 0, so 31 is prime.
The smallest P that does not result in a prime is 11; 2047 = 23 x 89.

Quote: RS

Life hack: Every prime number can be written as either 6n+1 or 6n-1.

Now, prove there are infinitely many primes of each form and I'll be impressed.

Quote: teliot

Now, prove there are infinitely many primes of each form and I'll be impressed.

If I have an hour or two of free time tomorrow then I'll knock out the proof real fast.




@TDG, that's a much better proof than what I was thinking....well, they basically both say the same thing, except yours is more elegant. Mine is more along the lines of:

1. Can't be an even number, because obvious. Only odd numbers remain eligible to be prime.
2. One third of odd numbers are divisible 3 (these are all 6n-3), making them ineligible. The other two-thirds of odd numbers fall on +/-1 from 6n if you look at a number line.


For fancy math lawyers, would you have to prove the point #2 further, or is that sufficient (particularly the second sentence)? My guess, in a formal proof, that yes you would have to.

TL:DR can you post the entire number

Quote: djatc

TL:DR can you post the entire number

Here it is (warning --about 1000 pages when it fully loads):

http://ijmp.org/m82589933/

I would prefer this number in base 2, but base 10 seems to be all the rage these days.

Is the last digit of this prime a 1,3,7 or 9? That's all I want to know. And, NO, I don't want to read through a 1000 pages to see how it ends.

It is 1.

Quote: LoquaciousMoFW

It is 1.

Base 2, almost all primes end in 1.

Quote: teliot

Base 2, almost all primes end in 1.

All but 10?

Quote: LoquaciousMoFW

All but 10?

All but 10?

All but 2?

You saying all except two primes end in a 1, eh?

Quote: RS

Life hack: Every prime number can be written as either 6n+1 or 6n-1.

This is a very stimulating observation, and ThatDonGuy has given a proof. I would be interested in seeing other proofs, possibly one based on the idea that such numbers both cannot be composite.

Sometimes only one or the other of these is prime and sometimes both are prime. These are called twin primes.

It suggests an easy method of generating primes, however it leaves open the question of which of the two numbers is prime.

It also suggests an easy method of primality testing: add 1 and subtract 1 from the number, and if either of these is divisible by 6, the number is prime, however there are some composite numbers that pass this test. In looking for such numbers I visited the Wolfram Fermat Pseudoprime page. While there I found some but also found an error. It gives 659 as a pseudoprime, however it is prime.

Perhaps there are other errors on the page.

Quote: netzer

Sometimes only one or the other of these is prime and sometimes both are prime. These are called twin primes.

It suggests an easy method of generating primes, however it leaves open the question of which of the two numbers is prime.

Tell me, which of 6 x 20 - 1 and 6 x 20 + 1 is prime? Hint: the first one is 7 x 17, and the second is 11 x 11.

In fact, for any positive integer N, you can find a set of N consecutive nonprime numbers:

Whether or not there is a largest N such that both 6N - 1 and 6N + 1 are prime is one of the Great Unsolved Problems Of Mathematics.

Quote: ThatDonGuy

Whether or not there is a largest N such that both 6N - 1 and 6N + 1 are prime is one of the Great Unsolved Problems Of Mathematics.

Not as unsolved as it used to be ... thanks to a professor at UCSB.

https://en.wikipedia.org/wiki/Yitang_Zhang

"On April 17, 2013, Zhang announced a proof that states there are infinitely many pairs of prime numbers that differ by 70 million or less. This result implies the existence of an infinitely repeatable prime 2-tuple,[2] thus establishing a theorem akin to the twin prime conjecture. Zhang's paper was accepted by Annals of Mathematics in early May 2013,[6] his first publication since his last paper in 2001.[15] The proof was refereed by leading experts in analytic number theory.[7] Zhang's result set off a flurry of activity in the field, such as the Polymath8 project."

ThatDonGuy has been eating his Wheaties! I think he has a degree in Computer Science from U. of Cal or Cal Tech. I wish people would put more information in their bios.

All right, here's the next one:

A large number has a remainder 1 when divided by 6. What is the probability that it is prime?