Problem 4

Prove that there is no function

f

from the set of non-negative integers into itself such that

f(f(n)) = n + 1987

for every non-negative integer

n

Step 4 of 5: Lift the fixed class

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In plain words

The fixed class means $f$ adds an integral number of 1987-blocks, and it adds those blocks twice under iteration.

f(a)=a+1987q,\quad q\in\mathbb Z_{\ge0},\qquad f(f(a))=a+2\cdot1987q

Detailed analysis

Choose $a\in\{0,\ldots,1986\}$ . The fixed-class relation gives $f(a)=a+1987q$ for some $q\in\mathbb Z_{\ge0}$ . Translation compatibility yields $f(f(a))=f(a+1987q)=a+2\cdot1987q$ .