Problem Wheels

The doubling wheel

Walk a number

Speed

Faint walks of every number up to

Color every number on the slide rule by how many moves it takes to reach 1

Seats

Rings Dim the seats that are settled

The walk

Chart

Proven and open

Proven: the share of unsettled seats keeps shrinking toward zero as the rings go out, so almost every number eventually drops below where it started. Each seat always splits into one climber and one faller, which is why the moves look like fair coin flips.

Checked: every number up to about 300 billion billion has been walked by computer and reaches 1.

Open: whether every number does. A number that never comes home would need a walk that climbs at least 63 moves in every hundred, on average, forever, or a loop other than 1 → 2 → 1. The settled share heading to zero only says such numbers are rare.

Zero waves

Waves

Zero wheels

Show up to

The wheels at x

Pretend zero

Move the first zero off the line, with its mirror twin How far off

Proven and open

Proven: with all the zeros, the waves rebuild the prime staircase exactly. That's Riemann's formula, finished by von Mangoldt. At least two fifths of the zeros are proven to sit on the line; a 2026 result, not yet peer reviewed, claims about two thirds.

Checked: every one of the first several trillion zeros sits on the line.

Open: the Riemann Hypothesis, that every zero sits on the line. Then every wave swells at the same slow rate, about √x, and the primes never stray from their average by more than that. One zero off the line would make a wave that outgrows all the others.

The folded wheel

Wrap the numbers from 0 to 2n once around a circle and fold it down the middle: number k lands face to face with 2n − k. Goldbach says some facing pair is always two primes. Gold chords join the pairs that are.

The even number

Show what wheel blocks

Why some even numbers have more pairs

Each small prime p is a wheel of p seats. A facing pair breaks if p divides either member, so p blocks two seats: k ≡ 0, and k ≡ 2n. When p divides 2n those two are the same seat, so the fold sends p's blocked seats onto themselves and more pairs survive.

Pairs for every even number

Each dot is one even number. They split into bands by which small wheels divide them, and the lines are what the wheels predict for each band. Tap the chart to jump there.

Proven and open

Checked: every even number up to 4 × 10¹⁸ is a sum of two primes.

Proven: every odd number above 5 is a sum of three primes (the weak Goldbach problem, settled in 2013). Every big enough even number is a prime plus a number with at most two prime factors (Chen).

Open: two primes, for every even number. The wheels predict dozens of pairs for numbers in the thousands and hundreds in the tens of thousands, so a failure would mean every one of them breaking at once.

Wheel seats

Each small prime is a gear with that many seats. A pattern like twin primes (n and n + 2) blocks the seats where one of its members is a multiple of that prime. As n steps along, every gear turns one seat, and n can only be a hit when every gear points at an open seat.

Pattern

Turn the gears

n

Speed

Counted against predicted

Proven and open

Proven: if some gear blocks every seat, the pattern dies after a few tiny cases. Some pair of primes no more than 246 apart turns up forever (Zhang, then Maynard and a big online collaboration).

Predicted: every pattern that no gear fully blocks turns up forever, at the rate its seats give. The counts match that rate to about a percent everywhere anyone has looked.

Open: every one of these. Nobody has proven there are infinitely many twin primes, Sophie Germain primes, or primes of the form n² + 1.

Prime races

Roll the primes onto a small circle and count how many land on each arm. In the long run the arms tie. Along the way, the arms where squares land almost always trail: every odd square lands on arm 1 of a circle of 4, and that small pile of squares is the head start the other arm gets.

Race

Also count each prime's square as half a prime, its cube as a third, and so on

The arms at x

Proven and open

Proven: in the long run every arm gets an equal share (Dirichlet, then the prime number theorem). The trailing arm takes the lead infinitely often, so the head start never settles the race for good (Littlewood).

Predicted: the swings around the head start are waves from the zeros of this circle's own zeta function, like the Riemann tab. If those zeros sit on their line too, and no two of them turn in lockstep, arm 3 of 4 leads 99.59% of the time, measured on a log scale (Rubinstein and Sarnak).

Open: the zeros being on their line, which this needs. Same hypothesis, one circle over.