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The Arithmetic Derivative (Part 4)

Updated: Oct 19, 2021

What do the differential equations mean?

Usually, we are motivated to study differential equations by situations in everyday life, nature, physics, and math. What motivates studying this arithmetic derivative?

The first answer is that it's a small adjustment to something much more commonly studied called a derivation in differential geometry. But focusing on the fact that we're defining D on the natural numbers, the real motivation is what it tells us about primes and numbers in general. In a sense, by setting it up so that D(p) = 1 for any prime p, we've DESIGNED the arithmetic derivative to tell us something about primes. So...what does it say?

Infinitude of primes

We showed in the second part of this series that if we assume we only have finitely many primes, then the derivative of their product is not divisible by any of those primes, so we must have more primes. Any finite list won't do, so we must have infinitely many primes.

Asymptotic number of primes

Once we started investigating the size of S(m) = {n | D(n) = m}, we found that for m > 1, the size of the solution set is finite and bounded by (m/2)^2. If m = 1, then S(m) is the set of primes numbers, is infinite. So it's useful to think of asymptotic size, meaning we give a growth rate for the size of S(1). The Prime Number Theorem tells us that this growth rate is N/log(N).

Twin Primes

A long-standing question in math is whether there exist infinitely many prime numbers p with p - 2 also prime. Though unsolved, a big breakthrough by Yitang Zhang in 2013 showed that there are infinitely many primes p with p - 70000000 also prime! My favorite fact is that the sum of the reciprocals of twin primes CONVERGES, despite the sum of reciprocals of all primes DIVERGING. To formulate twin primes with the arithmetic derivative, consider

If such an n exists, then D(D(n)) = 1, so D(n) = p for some prime p. Notice that if n = 2(p-2),

D(2(p-2)) = 2D(p-2) + (p-2)D(2)

If p is a twin prime, then p - 2 is prime, and D(p - 2) = 1, so

D(2(p-2)) = 2 + (p-2) = p

Great! So if p is a twin prime, then n = 2(p-2) satisfies our second-order differential equation.

If there are infinitely many twin primes, then the solution set is infinite, and we can again give asymptotics for its size, proportional to N/log(N)^2.

Irrationality of sqrt(2)

A number is called a rational number if it can be written as the ratio of two whole numbers. If it can't, then it's called irrational. A nice proof that appears in every discrete math course is proving that the square root of 2, sqrt(2), is irrational. We can prove this with the arithmetic derivative as well! It starts similarly as the usual method and is by contradiction:

Suppose that we had whole numbers a,b so that sqrt(2) = a/b, in reduced form, meaning a and b share no factors. Another way to say it is that a and b are coprime.

Rearranging gives

2b^2 = a^2

Now let's apply D,

D(2b^2) = D(a^2)

2D(b^2) + b^2 = 2aD(a)

Remember that a and b are coprime, meaning that at least one must be odd. The righthand-side 2aD(a) is clearly even, so since 2D(b^2) is even, this implies b^2 must be even as well.

This means b is even, so we can assume that a is the one that is odd. But the very first equation 2b^2 = a^2 implies that a^2 is even, which means a is even. A contradiction!

Thank you for reading!

Jonathan Gerhard


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