The Last Safe Positive Real Number

At a glance

The greatest positive real number in the Mandelbrot set is

At 𝑐 =14, the orbit is trapped in the safe interval [0,12]. For every 𝑐 =14 +𝑑 with 𝑑 >0, each step increases by at least 𝑑, so after sufficiently many steps the orbit is greater than 2 and escapes.

Key idea

The exact endpoint of this one-dimensional slice comes from two finite certificates. At the candidate endpoint, an invariant interval controls infinitely many iterates. Immediately to its right on the real axis, completing the square turns each step into a uniform positive drift. The first argument includes the endpoint; the second excludes every larger real number. It does not claim that the full complex set stops at this horizontal coordinate.

Worked solution, part by part

1. Build a safe zone at 𝑐 =14

Assume 0 ≤𝑧𝑛 ≤12. Then 0 ≤𝑧2𝑛 ≤14, so

In particular, 0 ≤𝑧𝑛+1 ≤12. Since the starting value 𝑧0 =0 is in [0,12], the same safe-step argument applies repeatedly. Every iterate remains in [0,12], so no absolute value can exceed 2. Therefore 𝑐 =14 is in the Mandelbrot set.

2. Complete the square to the right of the real endpoint

Let 𝑐 =14 +𝑑, where 𝑑 >0. Then

A real square is always nonnegative, so

3. Force an escape

Because 𝑧0 =0 and every step adds at least 𝑑, repeated addition gives

Choose any whole number 𝑛 >2𝑑. Then 𝑛𝑑 >2, so 𝑧𝑛 ≥𝑛𝑑 >2. Thus |𝑧𝑛| >2, and the orbit escapes. This argument works for every 𝑑 >0, however small.

4. Close both sides

Part 1 proves that 14 itself is included. Parts 2 and 3 prove that every larger positive real number has the form 14 +𝑑 for some 𝑑 >0 and is excluded. Consequently, 14 is exactly the greatest positive real point of the Mandelbrot set.

Proof and completeness

The conclusion needs both directions. Showing only that 𝑐 =14 is bounded would not rule out a larger value. Showing only that all 𝑐 >14 escape would not prove the endpoint is included. Together, the invariant interval and drift arguments establish both inclusion and maximality.

Scope check

This conclusion concerns parameters 𝑐 that are positive real numbers. The Mandelbrot set also contains nonreal points whose real parts are greater than 14, so 14 is not the easternmost real coordinate of the full two-dimensional set.

Check

For 𝑐 =14, the first few values 0,14,516,… are consistent with the interval bound but are not used as the proof. For 𝑑 >0, the inequality 𝑧𝑛 ≥𝑛𝑑 eventually crosses 2 because the Archimedean property guarantees a whole number larger than 2/𝑑.