MathLabs

Problem 6

Let pp be a prime number. Prove that there exists a prime number qq such that for every integer nn, the number np−pn^p-p is not divisible by qq.
Step 2 of 3: Compute the order
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In plain words

Compute the order

ord⁡q(p)=pandq=pk+1withp∤k\operatorname{ord}_q(p)=p \quad\text{and}\quad q=pk+1 \quad\text{with}\quad p\nmid k
Detailed analysis

We have pp≡1(modq)p^p\equiv1\pmod q. The order is not 11, since that would give q∣Φp(1)=pq\mid\Phi_p(1)=p; hence it is pp, so q=pk+1q=pk+1. The choice of qq gives p∤kp\nmid k.