MathLabs

Problem 2

Let n>6 be an integer and let a1,a2,…,aka_1,a_2,\ldots,a_k be all positive integers less than n and relatively prime to n, in increasing order. If a2−a1=a3−a2=⋯=ak−ak−1>0a_2-a_1=a_3-a_2=\cdots=a_k-a_{k-1}>0, prove that n is either a prime or a power of 2.
Step 1 of 5: Name the common gap
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d=a2−a1,a1=1d=a_2-a_1,\qquad a_1=1
Detailed analysis

The first reduced positive residue is a1=1. Let d be the common positive difference. Every listed residue is therefore 1 modulo d, and d divides the difference of any two listed residues.