Number Theory (Thailand Math POSN 3rd round)

Not so much interesting problems there, and I'm not allowed to take exams back.

Denote (an)\displaystyle \left(\frac{a}{n}\right) as Legendre symbol.

Def: Order of aa modulo nn denoted as ordn(a)ord_{n}(a) is the smallest positive integer kk such that

ak1(modn)a^{k} \equiv 1 \pmod{n}

Def: Zn={0,1,2,,n1}\mathbb{Z}_{n} = \{0,1,2,\dots,n-1\} be complete residue system modulo nn.

Def: Zn={xZn(x,n)=1}\mathbb{Z}_{n}^{*} = \{x \in \mathbb{Z}_{n} | (x,n) = 1\} be reduced residue system modulo nn.

1.) Find all positive integers nn such that

  • 2n302 \leq n \leq 30
  • For all xZnx \in \mathbb{Z}_{n}^{*}, ordn(x)<ϕ(n)ord_{n}(x) < \phi(n).

2.) Let aZpa \in \mathbb{Z}_{p}^{*} , pp be prime numbers. Prove that for all p17p \geq 17,

2.1) If (ap)=1\displaystyle \left(\frac{a}{p}\right) = 1, then ordp(a)<p1ord_{p}(a) < p-1

2.2) There exists aa such that (ap)=1\displaystyle \left(\frac{a}{p}\right) = -1 and ordp(a)<p1ord_{p}(a) < p-1

3.) Let pp be primes from 22 to 9797 inclusively. Find the number of primes such that

3.1) ordp(2)=p1ord_{p}(2) = p-1

3.2) (1p)=1\displaystyle \left(\frac{-1}{p}\right) = 1

Note by Samuraiwarm Tsunayoshi
4 years, 6 months ago

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