MathLabs

Problem 4

Prove that there is no function ff from the set of non-negative integers into itself such that f(f(n))=n+1987f(f(n)) = n + 1987 for every non-negative integer nn.
Step 3 of 5: An odd involution has a fixed class
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In plain words

An odd number of objects cannot be completely paired.

∣Z/1987Z∣=1987 odd  ⟹  g([a])=[a]|\mathbb Z/1987\mathbb Z|=1987\text{ odd}\implies g([a])=[a]
Detailed analysis

An involution pairs every non-fixed class with a distinct partner. Because there are 1987 classes, at least one class [a][a] is fixed.