Name: 78557 and Proth Primes - Numberphile
Uploaded: 2020-03-27T18:21:18.000Z
Duration: 8 min 40 s
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The superlative

If you look at the current top 10 in the prime charts, your see the top 10 largest primes at the moment.

Andi, you may notice that one of them stands out.

One of them is not quite like the others.

It was actually discovered fairly recently was discovered on October 31st 2016.

It was discovered on Halloween 2016 but all the other primes are Merced.

Send primes, all primes that are one less than a power off, too.

And there's a reason for that, because computational e it's quite easy to check if they are primes or not.

Right there is a There was a test for that which you can run quite quickly.

There's a couple of reasons why special on one of the reasons is that it takes is one step closer to solving a 50 year old Conjecture s.

So this number is called a profit frying.

So drop crimes have this form where it's okay, that's, um, number multiplied by a power to plus one.

One good do as take a value and I got a K is 90.

All right, let's take a is 19 and then we're gonna most play it by powers of to add one and they see we get a prime.

19 times 2 38 plus one equals 39 but that's not a crime.

Just check for another was trying 19 again.

So nine teams times for about 19 times two is 38.

We can double that again about 76 plus 1 77 last.

Both know that isn't 19 times two to the power for okay.

Looking to keep looking, I think that is plus one.

Let's keep looking for a prime as to the next 1 19 times 2 to 5 plus one.

That is a prime finally refund for Took awhile.

Yeah, it took a while, but eventually we found a prime in the 19 sixties, Mathematician said.

Pink Ski said that if you take something off this fall, it's K Times two to Power M plus one, and you keep running for it.

There are numbers that will never hit a prime, so you'll keep going forever and you'll never get it.

Probably family said they were infinitely many of them.

The natural question is, what's the smallest one?

So they think it might be this number 78,557.

So I'm saying if you multiply that by two to the end and then we add one we run through, that process will never hit a prime.

In fact, I know that's never gonna happen, so a mathematician called John Selfridge showed that that's never gonna happen.

He showed that this number, whatever it is, it will always be divisible by one of these 357 13 1937 or 73.

He said he's always gonna be divisible by one of those, so it's never going to be a prime.

Okay, this is a number that will never hit a prime.

But is it the smallest one that we don't know Now?

There's quite a few candidates for there's really In fact, we're going to check all the numbers smaller than 78,557.

Well, you can do that on most of those were eliminated until around about beginning of the 21st century.

17 hadn't been eliminated, so they might be numbers that are smaller than 78,557 on.

But thanks to advances in computation, that meant we could start eliminating these candidates now.

We started to eliminate them until they were six left These are the six that we left the smallest Waas.

So there were the six candidates left, but things started to slow down.

And I'm afraid there was a gap of nine years before we were able to eliminate one of these six.

And now this ISS struck after lest leaving five left.

Now, if we can eliminate those last five candidates, we will have proved that 78,557 is the smallest See Opinsky number.

Not so much found because they were searching for large primes because they were trying to eliminate case they That is part of the motivation for this This was found for by guy in Hungary on his home computer.

Where's downloaded a program You can get it from the prime grid website on this program was designed to eliminate those 17.

If I talked about on Dhe using that program, he finally eliminated 10,223.

So you can do this yourself using your own home computer so he gets the glory of solving out on dhe the glory off, striking off one of these candidates off our list.

So we have these candidates, and then you find a prime that kills officer candidate.

This one that was found recently is over nine million digits Long, massive number.

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78557 and Proth Primes - Numberphile

stuff

eliminate

vast

arbitrary