What remains open

What this section has proven, what is proven but would lead too far here, what nobody knows, and which statements of this website are interpretations. A map of the certain and the open.

At the end of this section a look back is worthwhile. Mathematics distinguishes strictly between what is proven and what is only conjectured, and this website tries to do the same. This chapter sorts the mathematical statements of the whole website into four groups.

Proven in this section

Everything in Chapters 1 to 10 is proven, most theorems also checked by Lean. Particularly important for the pages of the website are:

Statement	Chapter
$1 + 2 + \dots + n = \tfrac{n(n+1)}{2}$ , the tetractys is $T(4) = 10$	1
$1 + 3 + \dots + (2n-1) = n^2$ , the gnomon	2
The figure $\{n/k\}$ falls into $\gcd(n,k)$ parts, there are exactly four kinds	6
Ten is the first number with all four kinds	6
Primes are exactly the simplexes without a compound figure	6
Among composite numbers only 4, 6, 8, 9 and 12 have no compound star	6
Infinitely many primes, unique factorisation	4
Up to 24 all twin places are prime, the first exception is $25 = 5^2$	4, 5
Primes from 5 on lie on $6n \pm 1$ , and $24 \mid p^2 - 1$	5
$a^2 \equiv 1 \pmod m$ for all coprime $a$ exactly when $m \mid 24$	5
Only the decagon has as many full circles as corners	7
Angle sum equals quotient for only four figures	7
No stack of pure fifths gives pure octaves	8
$\sqrt{2}$ and $\varphi$ are not fractions	9
The folded squares tend to $\pi$	10

Beyond this: several observations that look like coincidences one by one have a common reason. The 10 and the 2.5 in the rasters of the sieve arise as $2 \cdot 5$ and $5 : 2$ , because 5 is the first number after 2 and 3 that neither of them divides. Some appearances of 24, such as the divisibility of $p^2 - 1$ , follow from Chapter 5, Theorem 4. Others, such as the kissing number in four dimensions or the Leech lattice, have reasons of their own.

Proven, but too far afield here

Some statements of the website are proven, only the proof is too long or needs tools this section does not introduce.

Statement	proven by
The share of coprime pairs tends to $6/\pi^2$	Dirichlet 1849, Mertens 1874, Cesàro 1881
$1^2 + 2^2 + \dots + n^2$ is a square only for $n = 1$ and $n = 24$ , $70^2$	Watson 1918
10, 120, 1540, 7140 are the only numbers above 1 that are both triangular and tetrahedral	Avanesov 1966
The kissing number in space is 12, in four dimensions 24	Schütte and van der Waerden 1953, Musin 2003
The densest sphere packings in 8 and 24 dimensions are $E_8$ and the Leech lattice	Viazovska 2016, with Cohn, Kumar, Miller, Radchenko
Kepler’s conjecture on the densest packing in space	Hales 2005, formally checked 2014
The primes spread evenly over the last digits 1, 3, 7, 9	Dirichlet 1837, de la Vallée Poussin 1896
There are about $x / \ln x$ primes below $x$	Hadamard and de la Vallée Poussin 1896
$\pi$ is not a fraction, and squaring the circle is impossible	Lambert 1761, Lindemann 1882

Even large proofs like these can be checked by computers today. Kepler’s conjecture was fully verified in proof assistants in 2014, published in 2017, and since 2024 a large international project in Lean has been working on Fermat’s Last Theorem.

What nobody knows

Some of the oldest questions about primes are open to this day, although they can be put in a single sentence.

The Riemann hypothesis (1859): all non-trivial zeros of the zeta function lie on the line with real part $\tfrac12$ . It would follow that the primes are distributed as evenly as is possible at all. It is one of the seven Millennium Problems, and since 2000 a prize of one million dollars has been offered for its solution (see The music of the primes).
The twin primes: are there infinitely many pairs $p, p + 2$ that are both prime? Yitang Zhang proved in 2013 that there are infinitely many pairs of primes less than 70 million apart. The bound has since fallen to 246. Whether it falls to 2, nobody knows.
Goldbach’s conjecture (1742, in this form from Euler): every even number above 2 is the sum of two primes. Checked up to $4 \cdot 10^{18}$ , not proven.
Primes of the form $n^2 + 1$ : are there infinitely many? In 1912 Edmund Landau named the question among four problems he considered unassailable. All four are still open, the twin primes and Goldbach among them.

Zeros

The explicit formula of Riemann and von Mangoldt builds the staircase of the primes from a straight line and one wave for each zero of the zeta function. With more zeros the staircase gets sharper. That all these zeros lie on one line is the Riemann hypothesis.

The observation that the spacings of the zeros follow the same statistics as the energy levels of heavy atomic nuclei is also a finding without proof. Hugh Montgomery and Freeman Dyson noticed the connection in 1972, Andrew Odlyzko confirmed it on billions of zeros. Why it holds is open.

Interpretations

And finally there are statements on this website that are not mathematics in the strict sense but readings. They are formulated there as theses, and that is how they should be read:

that by the tetractys the Pythagoreans meant exactly these connections between figure and divisibility,
that a “decimal coding” underlies the distribution of the primes. What is mathematically established is that the 10 arises there as $2 \cdot 5$ , independent of the number system but dependent on the choice of the rhythm of six,
that the simplexes express the essence of the circle number. The circle number appears in divisibility, as $6/\pi^2$ , and in the polygons, as the limit of Archimedes, but by two different routes that, as far as we know today, are not connected,
and all the connections to mysticism, physics and philosophy that stand in the other sections.

Interpretations are no flaw. Without them there would be no questions, and without questions no proofs. The Pythagoreans too pursued their mathematics out of an interpretation of the world, which Aristotle put into the formula that all is number. The mathematics that grew from it has proven them right in much and contradicted them in some things, and the incommensurable of Chapter 9 in particular shows that it stayed honest in doing so.

Sources and further reading

Sources