A prime number that looks like Gauss

Ajouté : 2026-05-12

[00:00:00]A prime number that looks like Gauss himself. I saw this post and immediately I got very suspicious. Did they really check that the number that is making this image is a prime number? And a spoiler alert, no, they did not check that this is an actual prime number.

[00:00:19]Although we're pretty darn sure that it is a prime number. Let me explain. Let me first show you what the entire image looks like. It's some very large number.

[00:00:30]It does look like Gauss, but it's kind of blurry. So the first thing I did was go to ChatGPT, extract the number in the image, which by the way it has some 4,080 digits, and reconstruct the image again from this actual number so that I have a sharper image with sharper digits. With another internet search, I did find the person who did this in the first place, and the explanation of how they came up with this image and this number that they claim to be prime that looks like Gauss. The first step is that if you have any image, there is some software that allows you to create an array of numbers that looks exactly like that image. And the reason why I was suspicious that this number had been checked to be prime is that it is about 8 * 10 ^ 4,079, and that is a very large number. It is not of the form of a Mersenne number or a Fermat number, and therefore to check the primality of a number that big, it would take some serious computing power, and I don't think they would have done that much computing for something like this. The question is, how do you choose a number that looks like Gauss and it is a prime number? And the key to that is in this corner of the image. If you zoom in and then you look at the very first digits of the number, then you see that there is some array of numbers here that do not seem to add to the actual picture. So, those numbers, the first few numbers, have been chosen. Those are the ones that have been modified so that the entire number is a prime number.

[00:02:11]But, what are the chances that you're going to be able to modify the first few digits of the number and get a prime number? Well, the prime number theorem tells me that a number n is a prime with probability 1 over log of n. Since this number is about 8 * 10 to the 4,079, then the probability that this number is prime is about 1 in 10,000. So, what they did is come up with a number that looks like Gauss and then run through the next 10,000 numbers and see if any of those is a prime number. Now, the problem is that a number this big is very hard to verify that is an actual prime number.

[00:02:53]So, instead of proving that a number is prime, what they did is use a probabilistic test. In this case, the Baillie-PSW probability test. Which is so accurate that we haven't found a single composite number that passes the Baillie-PSW test.

[00:03:13]So, then again, what you do is you find a number that looks like Gauss, then run over the next 10 or 20,000 numbers. For each one of those numbers, you run the Baillie-PSW test until you find one of them that does pass the test. And that's how they found this number, which is very likely a prime number, but they have not actually proved that it is prime.