Largest prime number?

Have you already read the page named Largest numbers? If not, are you familiar with reductio ad absurdum? It's not about anything absurd! It's all about using contradiction for a better purpose.

Here we will prove a wondrous result from number theory, an old and venerable field of mathematics. But, the point is that you won't need to appeal to authority, anymore. You can be the authority, if you choose, and say with confidence that you know that there are infinitely many primes! By the way, do you know what it means to say "infinitely many?" Well, soon, thanks to doing this proof yourself, you will. You are probably familiar with prime numbers. But, if not, that's OK. They are not hard to get to know. Here is how we can define them: a number is prime if it is greater than 1 and if it cannot be divided evenly, meaning without a remainder, except by 1 and itself. For example, 7 is a prime. Dividing it by 2 gives a remainder of 1. 7 over 3 and 7 over 5 also give remainders: 1 and 2, respectively. It's not divisible by any other number, but, 7 and 1. 

Ok, how do we do this? Well, just as we did in the short essay on largest numbers, we will use proof by contradiction: Let's assume that there is a number P which is the largest or greatest prime number. Of course, we don't know that. That could be false. But, that's why we said let's assume. Trust me that it will help us find out the truth of the matter.

Now that we have a simple assumption, let's use it: Let's find all the prime numbers less than P and multiply them all together with P. E.g., if P were 7, we would do the following: 2 x 3 x 5 x 7 + 1. That simple equation gives us 211. That's a prime! If you don't believe me just ask Google (not ChatGPT, that smart large language model is not as wise as knowing about prime numbers yet!) And, by the way a crucial point here: Please don't you ever believe me (or anything else :-)! Never believe anything I say or write here. Make sure to check for yourself.

At this stage, it might help to do a few examples by hand: 2 + 1 is 3 and that's a prime. Check. 2 x 3 + 1 is 7 and it's another prime. Check, check. Oh, wait, also notice that 7 is not the next prime in the series. This method doesn't generate all primes. It skips over some. That's good to know. And you, too, do know! But, again, don't believe me. Check for yourself. You do know, don't you?

This gives us some confidence that we can get bigger and bigger primes by just doing what we did. Right? But, remember that confidence is not knowledge and we don't have the truth yet! But, we are close. Are we? Can you tell? Before going further, just give it some thought. And no sweat. Here, there is no threat! We could also generalize that nothing real can be threatened. Ever. But, that could be a much longer story. My younger son, when he was only about 8, insisted that nothing unreal can be threatened. And, he was right, too. I always learn so much from kids. Don't you?

Back to this story.. We can do more multiplications and get more results like 2x3x5x7x11+1. We need to make sure we don't miss any primes, but notice that we didn't multiply with 9. We don't need it. Why not? (I won't answer this one. It's easy. Know for yourself! But here is an important parenthetical: check to see what you feel when you attempt to know something for yourself. Does that give you pain or pleasure? You will be getting to know yourself, too! Isn't that amazing?? But, I went too far and misspoke. What you are getting to know is math and your mind. Not your self! How can one know oneself? Is that even possible? I posit not! But, don't believe me. Check for yourself. This is a great and immortal metaphysical question. You are at the frontier!) 

Ok, I assume you did some soul searching, strike that, some mind searching, and realized that there is something cool happening here. We assumed that the largest prime number is P. But, we got an even larger number by multiplying all the smaller prime numbers and P together and adding one. And that number, which we will denote it as L, is likely to be another prime. If it is, then, we are done. QED. But, are you sure that's the case? 

How about 2x3x5x7x11x13 + 1? Do you know that it's a prime? Ask google! Do! You'll find out that it is 30031 and it is not a prime! Are you surprised? I hope you are not upset. But, surprise is a good thing. It is informative. 30031 (or, the number L if P were 13) is not a prime, but, we do know that it is not divisible by 2, 3, 5, 7, 11 and 13. Why not? Because the division would give the remainder of 1. That's why we added 1. 

Well, if there is still some suspense, let's roll down the curtain (the play is almost over): we didn't find a prime number larger than 13, but we know that it must exist! How come? Well, we know that 30031 is not a prime. Google tells us. And we can check. Ask google the following question: "prime factors of 30031." It will tell you that they are 59 and 509. Cute, isn't it? 59 * 509 indeed is 30031 and you can easily do the math if you don't trust Google :-).  An easier way for me is to do the following 60 x 509 = 6 x 5090 = 3 x 2 x 5090 = 3 x 10180. But, don't forget to subtract 509. 30000 + 3 x 180 - 509 = 30k + 300 + 240 - 509 = 30031, yes!

So, L = 30031 is not a prime, but we found not one but two primes bigger than 13. And, that does indeed complete our proof that no matter how big a prime number we got, we can always do, at least in principle, what we did here and know that that would give us either one or more primes greater. It is funny we know that such big primes exists, even if we can't see them yet, as we don't know their values. See this wiki page if you are also intrigued by this. 

Our proof is done. Spend a few minutes, go over what we did above a few times and please convince yourself. This is very good news. You demonstrated to yourself that you can get to know things truly by making an assumption and then checking whether you can get a contradiction or not. If you get a contradiction, great! You got new knowledge! If not, you have a bit of confidence that your assumption might be true. But, if I were you, I wouldn't believe it. It might after all be just a false assumption. Hereby I posit, most assumptions we make are false. But, don't believe me. Check for yourself. And as Wittgenstein said, please don't be encumbered by questions (is this assumption true or false? How about that assumption? How about the other? Ad nauseam!). Just take them one at a time. Now, we get to a very personal question: which assumption would you take on? I leave you to it. Be still. Peace.

PS. If you program the computer, or if you can read code, check out a program I wrote to demonstrate this proof with more examples here: Scala is Fun! 

PPS. While we are at it, you can learn to program, too! Try an online graphical learning environment that uses cool computer graphics, animations and games all written with Scala and Java: Kojo is even more Fun!