MathLabs

Problem 5

Let N={1,2,3,…}\mathbb{N}=\{1,2,3,\ldots\}. Determine whether there exists a strictly increasing function f:N↦Nf:\mathbb{N}\mapsto\mathbb{N} such that (i) f(1)=2f(1)=2; (ii) f(f(n))=f(n)+nf(f(n))=f(n)+n, (n∈N)(n\in\mathbb{N}).
Step 3 of 5: Prepare the composition
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In plain words

The quadratic relation makes the error term shrink into exactly one unit interval.

φ2=φ+1,φ(φ−1)=1\varphi^2=\varphi+1,\qquad \varphi(\varphi-1)=1
Detailed analysis

For a fixed n, write the nonintegral quantity inside the floor as an integer m plus a fractional part delta. Irrationality of phi makes delta strictly between 0 and 1. Use the two displayed golden-ratio identities to expand the next application of f.