Are there "infinitely many prime numbers"?
Modern encryption technology relies on prime numbers. Prime numbers have no regularity, and it is difficult to determine whether a large number is prime. Even if you use a supercomputer to try to find the divisors of a large number, you cannot actually find them in reality. Therefore, even if communication is illegally intercepted, there is no need to worry about the encryption being broken.
By the way, are there infinitely many prime numbers? In other words, is there a largest prime number? If there is a "largest prime number," it means that the number of prime numbers is finite. If there is "no largest prime number," it means that there are infinitely many prime numbers.
Now, let's prove this properly. Well, here is the [Problem].
[Question] To show that "there is no largest prime number," the following argument was made.
Assume there is a largest prime number, and let that number be N.
Here, let M be the number obtained by adding 1 to the product of all prime numbers less than or equal to N.
That is, M = 2 × 3 × ... × N + 1.
At this time, M[ A ]N.
Also, no matter what prime number from 2 to N you divide M by, it will always leave a[ B ]remainder.
Therefore, M is a prime number.
However, this contradicts the fact that "N is the largest prime number."
From the above, it has been shown by proof by contradiction that "there is no largest prime number."//
(1) Insert the appropriate equality or inequality sign into [ A ], and the appropriate number into [ B ].[ A ][ B ](2) If this proof is correct, write "⭕️" in the answer box. If the conclusion itself is wrong, write "❌". If there is a flaw in the proof, rewrite only one line out of the eight lines of the proof above to correct it.
only
This proof is included in "Euclid's Elements," written by the ancient Greek mathematician Euclid. It is a book written around the 3rd century BC. I arranged it into an exam question.
Now, let's move on to the [Answers and Explanations].
(1) From M = 2 × 3 × ... × N + 1, first, M > N holds.
Next, if you divide M by 2, the remainder is 1. If you divide M by 3, the remainder is 1.
Similarly, no matter what prime number less than or equal to N you divide M by, the remainder is always 1.
Therefore, the answer is[ A ] is >, and [ B ] is [ B ]1.
(2) Now, as for whether we can say from this that "M is a prime number larger than N," unfortunately, we cannot say that.
From (1), "M is not divisible by any prime number less than or equal to N," but it "might be divisible by a prime number larger than N."
And in either case, it means "there is a prime number larger than N," which contradicts "N is the largest prime number," so in the end, it is proven that "there is no largest prime number," or in other words, "there are infinitely many prime numbers."
For example, "when N = 13, M = 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30031 = 59 × 509," so M itself is not a prime number, but in this case too, there are prime numbers (59 and 509) larger than N = 13.
Therefore, the answer to (2) is as follows (↓).
It is complete if you rewrite the 6th line "M is a prime number" to "M is a prime number or a composite number divisible by a prime number larger than N".
By the way, can you find the divisors of 30031 on your own? Well, that would be difficult. I found the example above on the internet and posted it here, but I gave up trying to find an example on my own. To return to the previous topic, the security of encryption is guaranteed precisely because of that difficulty.
I gave the [Problem] above in the end-of-year exam at the school where I work (2026.03.07). I did a " proof problem that does not state the conclusion " in the third-term class, and this question was based on that. It is a math exam for second-year high school students (liberal arts track).
Incidentally, the header image of this article is Imam Square in Isfahan, Iran. Iran is a very beautiful place. Please take a look here if you like.