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A power digraph modulo $n$ , denoted by $G (n, k)$ , is a directed graph with $Z_{n} = {0, 1, \dots, n - 1}$ as the set of vertices and $E = {(a, b) : a^{k} \equiv b (mod n)}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G (n, k)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G (n, k)$ contains exactly two components. The primality of Fermat number is also discussed.

Complete solution of the Diophantine equation $x^{y} + y^{x} = z^{z}$

Mihai Cipu (2019)

Czechoslovak Mathematical Journal

The triples $(x, y, z) = (1, z^{z} - 1, z)$ , $(x, y, z) = (z^{z} - 1, 1, z)$ , where $z \in ℕ$ , satisfy the equation $x^{y} + y^{x} = z^{z}$ . In this paper it is shown that the same equation has no integer solution with $min {x, y, z} > 1$ , thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Zhi-Hong Sun (2011)

Acta Arithmetica

Congruences for $q^{[p / 8]} (m o d p)$

Zhi-Hong Sun (2013)

Acta Arithmetica

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p / 8]} (m o d p)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.

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An application of a formula of Western to the evaluation of certain Jacobsthal sums

An application of dihedral fields to representations of primes by binary quadratic forms

An Artin Character and Representations of Primes by Binary Quadratic Forms III.

Applications of Jacobi's symbol to Lehmer's numbers

Arithmetischer Beweis des Reciprocitätsgesetzes für die biquadratischen Reste.

Bemerkungen zum analytischen Beweise des cubischen Reciprocitätsgesetzes

Bestimmung des quadratischen Rest- Charakters durch Kettenbruchdivision. Versuch einer Ergänzung zum Dritten und fünften Beweise des Gauß'schen Fundamental- Theorems

Beweis des Reciprocitätengesetzes für die quadratischen Reste. (Aus eine Aufsatze in No. XXIII der Sitzungsberichte der Berliner Akademie 1884).

Beweis des Reziprozitatsgesetzes für quadratische Reste

Binomial coefficient predictors.

Characterization of power digraphs modulo $n$

Complete solution of the Diophantine equation $x^{y} + y^{x} = z^{z}$

Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Congruences for $q^{[p / 8]} (m o d p)$

Congruences for representations of primes by binary quadratic forms

Connection between the Wieferich congruence and divisibility of h⁺

Contribution à la théorie des résidus cubiques et biquadratiques

Contribution à la théorie des résidus cubiques et biquadratiques

Criterion for 2 to be an lth power

Das spezielle Reziprozitätsgesetz im relativbiquadratischen Zahlkörper

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