Dreaming of prime numbers in short intervals

I still remember Kaisa telling in primary school, that Euclid's proof of infinite number of prime numbers is often considered as the most beautiful proof in mathematics.

I like many of quanta&#x27;s articles on mathematicians and scientists, and I appreciate their contributions to science journalism. I would like to add one aspect that I think this article downplays, but which is understandable to a large audience.<p>Kaisa and Maksym study multiplicative functions, i.e. functions which satisfy f(ab) = f(a)f(b) if a and b are relatively prime. A big part in Kaisa and Maksym&#x27;s fundamental technique boils down to understanding completely the behavior of f on small primes, and giving somewhat loose bounds on the rest. Central to their success is their ability to make quantitative the intuitive statement that &quot;most numbers have lots of small primes as factors&quot;. This required a few new ideas, but the nugget is quite simple, I think.

As a layman, I wonder about the connection between primes and random numbers. Obviously, the primes are not actually random, but they seem to exhibit a lot similarity to random numbers in that they are chaotic over short intervals, but seem smoother and better behaved over long intervals.

Can anyone explain to me why primes are so important in cryptography?<p>I understand that they are used for factorization, where it is easy to multiply two primes and get a number, but it is difficult from a big number to find what two primes were used to get it. So, the question is why not to have a big database of primes, and when we have that big factorized number we try to divide that number by each prime from the database and see if the result is another prime from the database? Wouldn&#x27;t that work?

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Their example uses log_10 instead of ln