Andrica's Conjecture

I've always loved to stare at prime numbers. Probably that happens because I really do not understand their properties. However, taking into consideration the limited capacity of my own skull, I like at least to try to understand what others, much more capable than me, have to say about prime numbers.

And taking about others, I tried these days to find what other Romanian fellows have to say about this topic. And one interesting thing I found is called Andrica's Conjecture.

Andrica's Conjecture

Prof. Univ. Dr. Dorin Andrica works at the Babeş-Bolyai University, Cluj, Romania where he is a full professor within the Faculty of Mathematics and Computer Science.

This is a list of books from amazon.com which contains his name: list.

Among them, there is a book co-authored by the regretted Ioan Cucurezeanu, which was also my own Arithmetics professor. My only professor who put me to take the same exam nine times in a row (for the whole sumer holiday).

Andrica's Conjecture definition can be found here: http://planetmath.org/AndricasConjecture.

It simply states that: Given the n-th prime \(p_{n}\), it is always the case that \(1 > \sqrt{p_{n+1}}-\sqrt{p_{n}}\).

The conjecture is unproved until now. The description states that the conjecture has been proven by computers up to n = 10^5. This looks odd since the vector of primes which I have is build with a trivial version of a sieve and has 50847534 prime numbers.

As always I try to use my rapaio tools, so plotting a line with the Andrica's gap function is easy like

    private void drawAndrica(Vector primes, int size) {
        Frame m = Frames.newMatrixFrame(size, new String[]{"index", "gap"});
        for (int i = 0; i < m.getRowCount(); i++) {
            m.setValue(i, "index", i + 1);
            m.setValue(i, "gap", sqrt(primes.getValue(i + 1)) - sqrt(primes.getValue(i)));
        }
        draw(new Plot()
                .add(new Lines(m.getCol("index"), m.getCol("gap")))
                .add(new ABLine(0, true))
                .setYRange(0, 0.7),
                700, 300);
    }

First we plot the function for the first 500 values.

What is already been discovered is that the largest known difference in this function happens for a small n, aka for n=4.

The plotted function looks like a decreasing function (not monotone, but decreasing on average). The question is, if we plot many values the tendency remains the same?

This are all the prime numbers I have at hand. There are more than 50 millions. And the graph shows the tendency is to decrease on average. Which is really interesting. This is not a proof, of course, but one could find reasonable to expect the conjecture to be true. Nice work Dr. Andrica!