numerunique's style
09/11/2025

A self-certifying program

The octaphi converter presented in a previous post is able to convert any natural integer in a sum of powers of φ while φ is an irrational roughly equal to 1.62…

Try to compute ${\phi }^6+{\phi }^2+{\phi }^0+{\phi }^{-2}+{\phi }^{-6}$ taking φ = 1.62, you'll find 22.14.

It is far from trivial that a sum of powers of an irrational can be equal to an integer.

The possibility that would work is due to the magical properties of φ: ${\phi }^2=\phi +1$ and ${\phi }^{-1}=\phi -1$ (properties that are accessible to a mathematical level as low as knowing the square root operation).

However, the octaphi converter, thanks to Burnol's algorithm, works with only exact computations on integers. Thus all operations must be perfect and we have here the case of a very rare self-certifying program!

Less than the visible tip of an iceberg

What do you see in the octaphi converter?

Just an input field that filters decimal digits and instantly converts any number (arbitrarily up to ${\mathrm{10}}^{128}-1$ ) to its equivalent in a sum of powers of φ.

Yet the process behind involves complex computations with no approximation. You see less than the tip of an iceberg and it simply works.

Now, at last, the octaphi converter usefulness is visible. It illustrates numerunique's style: a basic user interface hiding a complex process that works flawlessly.

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