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Why quintic polynomial equations are not solvable in radicals

Křížek, Michal;  Somer, Lawrence — 2015

Application of Mathematics 2015

We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations + , - , · , : , and · n . Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.

Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

Walter Carlip;  Lawrence Somer — 2007

Colloquium Mathematicae

Let d be a fixed positive integer. A Lucas d-pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U(P,Q) such that the rank of appearance of N in U(P,Q) is exactly (N-ε(N))/d, where the signature ε(N) = (D/N) is given by the Jacobi symbol with respect to the discriminant D of U. A Lucas d-pseudoprime N is a primitive Lucas d-pseudoprime if (N-ε(N))/d is the maximal rank of N among Lucas sequences U(P,Q) that exhibit N as a Lucas pseudoprime. We derive...

On a connection of number theory with graph theory

Lawrence Somer;  Michal Křížek — 2004

Czechoslovak Mathematical Journal

We assign to each positive integer n a digraph whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a 2 ≡ b ( m o d n ) . We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.

On semiregular digraphs of the congruence x k ≡ y ( mod n )

Lawrence Somer;  Michal Křížek — 2007

Commentationes Mathematicae Universitatis Carolinae

We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . The digraph G ( n , k ) is semiregular if there exists a positive integer d such that each vertex of the digraph has indegree d or 0. Generalizing earlier results of the authors for the case in which k = 2 , we characterize all semiregular digraphs G ( n , k ) when k ≥ 2 is arbitrary.

Square-free Lucas d -pseudoprimes and Carmichael-Lucas numbers

Walter Carlip;  Lawrence Somer — 2007

Czechoslovak Mathematical Journal

Let d be a fixed positive integer. A Lucas d -pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U ( P , Q ) such that the rank of N in U ( P , Q ) is exactly ( N - ε ( N ) ) / d , where ε is the signature of U ( P , Q ) . We prove here that all but a finite number of Lucas d -pseudoprimes are square free. We also prove that all but a finite number of Lucas d -pseudoprimes are Carmichael-Lucas numbers.

The structure of digraphs associated with the congruence x k ≡ y ( mod n )

Lawrence Somer;  Michal Křížek — 2011

Czechoslovak Mathematical Journal

We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . We investigate the structure of G ( n , k ) . In particular, upper bounds are given for the longest cycle in G ( n , k ) . We find subdigraphs of G ( n , k ) , called fundamental constituents of G ( n , k ) , for which all trees attached to cycle vertices are isomorphic.

Bounds for frequencies of residues of second-order recurrences modulo p r

Walter Carlip;  Lawrence Somer — 2007

Mathematica Bohemica

The authors examine the frequency distribution of second-order recurrence sequences that are not p -regular, for an odd prime p , and apply their results to compute bounds for the frequencies of p -singular elements of p -regular second-order recurrences modulo powers of the prime p . The authors’ results have application to the p -stability of second-order recurrence sequences.

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