Computer assistance and pure mathematics

Today I want to tell you a story of a preprint in pure mathematics that came into existence only by crucial help of precise computer computations.

To explain the results, let us first define for a set \(A \subset \mathbb{N}_{>1}\) of natural numbers \[f(A) := \sum_{n \in A} \frac{1}{n \log(n)}\,.\] For \(k \ge 1\) let us further define \(\mathbb{N}_k\) as the set of those natural numbers with exactly \(k\) prime factors (counted with multiplicity); note that \(\mathbb{N}_1\) is the set of prime numbers.

  • 1993 it was proved by Zh. Zhang (DOI:10.2307/2153122) that \(f(\mathbb{N}_k) < f(\mathbb{N}_1)\) for every \(k > 1\).
  • 2013 it was conjectured by W. D. Banks and G. Martin (arXiv:1301.0948) that \(f(\mathbb{N}_k) < f(\mathbb{N}_{k-1})\) for every \(k > 1\).
  • 2016 a special case of the conjecture was proved by J. Bayless, P. Kinlaw and D. Klyve (DOI:10.1090mcom/3416), namely that \(f(\mathbb{N}_3) < f(\mathbb{N}_2)\).

Now in the preprint that I want to tell you about (arXiv:1909.00804) the author J. D. Lichtman computed precisely the values of \(f(\mathbb{N}_k)\) for \(2 \le k \le 10\) up to 20 digits, and got the following plot:

As we see, the data shows that \(f(\mathbb{N}_k)\) actually

  1. has a strict minimum at \(f(\mathbb{N}_6)\),
  2. decreases monotonically until \(k=6\) and then increases monotonically, and
  3. converges towards \(1\) as \(k \to \infty\).

J. D. Lichtman then managed to prove Points 1. for all \(k \in \mathbb{N}_{>1}\) and he also proved Point 3. But note that his computations of the values up to 20 digits are precise, which means that Point 2. is also proven for the values of \(k\) up to 10. So the only remaining open statement from above is that \(f(\mathbb{N}_k)\) increases monotically for \(k \ge 10\).

Especially, his computations disproved the conjecture of Banks and Martin.

All the above information is taken from the page The Erdös primitive set conjecture, where you can also find information about related problems about (prime) numbers.

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