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Number Theory (Thailand Math POSN 3rd round)

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

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

Def: Order of \(a\) modulo \(n\) denoted as \(ord_{n}(a)\) is the smallest positive integer \(k\) such that

\[a^{k} \equiv 1 \pmod{n}\]

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

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

1.) Find all positive integers \(n\) such that

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

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

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

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

3.) Let \(p\) be primes from \(2\) to \(97\) inclusively. Find the number of primes such that

3.1) \(ord_{p}(2) = p-1\)

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

Note by Samuraiwarm Tsunayoshi
2 years, 4 months ago

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