Lehmer Numbers Research Articles

Partitions into two Lehmer numbers

We prove an asymptotic formula for the number of representations of an integer as sum of two Lehmer numbers which implies that under some natural restrictions each sufficiently large integer is a sum of two such numbers.

Classifying real Lehmer triples: a revived computation

In this article we build on the work of Schinzel \cite{schinzelI}, and prove that if $n>4$, $n\neq 6$, $n/(\eta \kappa)$ is an odd integer, and the triple $(n,\alpha,\beta)$, in case $(\alpha-\beta)^2>0$, is not equivalent to a triple $(n,\alpha,\beta)$ from an explicit table, then the $n$th Lehmer number $u_n(\alpha, \beta)$ has at least two primitive divisors.

Partitions involving D.H. Lehmer numbers

Let n ≥ 2 be a fixed integer, let q and c be two integers with q > n and (n, q) = (c, q) = 1. For every positive integer a which is coprime with q we denote by \({\overline{a}_{c}}\) the unique integer satisfying \({1\leq\overline{a}_{c} \leq{q}}\) and \({a\overline{a}_{c} \equiv{c}({\rm mod}\, q)}\). Put $$L(q)=\{a\in{Z^{+}}: (a,q)=1, n {\not\hskip0.1mm|} a+\overline{a}_{c} \}.$$ The elements of L(q) are called D.H. Lehmer numbers. The main purpose of this paper is to prove that (under natural conditions) every sufficiently large integer can be expressed as the sum of three D.H. Lehmer numbers.

Two Diophantine approaches to the irreducibility of certain trinomials

In the previous paper [FLSU], we determined the Galois groups associated with some polynomials constructed through the use of circulant matrices. In the process of determining the Galois groups, the irreducibility of the trinomial x + x +m was established, where p represents an odd prime and m an integer ≥ 2. The approach there was based on a method of Lebesgue [Le]. In this paper, we discuss two other approaches we discovered in our investigations, both relying on the nice work of Bilu, Hanrot, and Voutier [BHV] (with an appendix by Mignotte) on Lucas and Lehmer numbers. In the next section, we consider the more general polynomials tp(x) = x +bx+c where b and c are nonzero integers. There are four cases where reducibility is easily established:

New pseudorandom sequences constructed by quadratic residues and Lehmer numbers

Let p p be an odd prime. Define \[ e n = { ( − 1 ) n + n ¯ , a m p ; if n is a quadratic residue mod p , ( − 1 ) n + n ¯ + 1 , a m p ; if n is a quadratic nonresidue mod p , e_n = \begin {cases} (-1)^{n+\overline {n}}, & \text {if $n$ is a quadratic residue mod $p$},\\ (-1)^{n+\overline {n}+1}, & \text {if $n$ is a quadratic nonresidue mod $p$}, \end {cases} \] where n ¯ \overline {n} is the multiplicative inverse of n n modulo p p such that 1 ≤ n ¯ ≤ p − 1 1\leq \overline {n}\leq p-1 . This paper shows that the sequence { e n } \{e_n\} is a “good" pseudorandom sequence, by using the properties of exponential sums, character sums, Kloosterman sums and mean value theorems of Dirichlet L L -functions.

On a Problem of D. H. Lehmer

The main purpose of this paper is to use the Fourier expansion for character sums and the mean value theorem of Dirichlet L–functions to study the asymptotic property of the difference between a D. H. Lehmer number and its inverse modulo p (an odd prime). A interesting mean square value formula is also given.

On groups generated by two positive multi-twists: Teichmueller curves and Lehmer’s number

From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these determine the same set of Teichmueller curves as the non-obtuse lattice triangles which were classified by Kenyon, Smillie, and Puchta. We also identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and show that this is minimal for the groups under consideration. In addition, we describe a connection to work of McMullen on Coxeter groups and related work of Hironaka on a construction of an interesting class of fibered links.

The Multiplicative Group Generated by the Lehmer Numbers

The Multiplicative Group Generated by the Lehmer Numbers

Existence of primitive divisors of Lucas and Lehmer numbers

We prove that for~${n>30}$, every~$n$-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

On systems of composite Lehmer numbers with prime indices

On systems of composite Lehmer numbers with prime indices

On the difference between a D. H. Lehmer number and its inverse modulo q

On the difference between a D. H. Lehmer number and its inverse modulo q

The G.C.D. in Lucas Sequences and Lehmer Number Sequences

The G.C.D. in Lucas Sequences and Lehmer Number Sequences

Lehmer numbers and an asymptotic formula for π

Let { a n } n = 0 ∞ be an integer sequence defined by the non-degenerate binary linear recurrence a n = A a n − 1 + Ba n − 2 , where a 0 = 0, a 1 ≠ 0, and A, B are fixed non-zero integers. It is proved, for a certain constant κ, that 6(1−K) log|a 1a 2…a N| log[A 1,a 2,…,a N] 1 2 =π+0 1 logn , which is the generalization of the formula of P. Kiss and F. Mátyás.

Applications of Jacobi's symbol to Lehmer's numbers

Applications of Jacobi's symbol to Lehmer's numbers

On some arithmetical properties of Lucas and Lehmer numbers

On some arithmetical properties of Lucas and Lehmer numbers

On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with parameters 𝐿, 𝑄 in arithmetic progressions

Let U n = ( α n − β n ) / ( α − β ) {U_n} = ({\alpha ^n} - {\beta ^n})/(\alpha - \beta ) for n odd and U n = ( α n − β n ) / ( α 2 − β 2 ) {U_n} = ({\alpha ^n} - {\beta ^n})/({\alpha ^2} - {\beta ^2}) for even n, where α \alpha and β \beta are distinct roots of the trinomial f ( z ) = z 2 − L z + Q f(z) = {z^2} - \sqrt L z + Q and L > 0 L > 0 and Q are rational integers. U n {U_n} is the nth Lehmer number connected with f ( z ) f(z) . Let V n = ( α n + β n ) / ( α + β ) {V_n} = ({\alpha ^n} + {\beta ^n})/(\alpha + \beta ) for n odd, and V n = α n + β n {V_n} = {\alpha ^n} + {\beta ^n} for n even denote the nth term of the associated recurring sequence. An odd composite number n is a strong Lehmer pseudoprime with parameters L, Q (or slepsp ( L , Q ) {\text {slepsp}}(L,Q) ) if ( n , D Q ) = 1 (n,DQ) = 1 , where D = L − 4 Q ≠ 0 D = L - 4Q \ne 0 , and with δ ( n ) = n − ( D L / n ) = d ⋅ 2 s \delta (n) = n - (DL/n) = d \cdot {2^s} , d odd, where ( D L / n ) (DL/n) is the Jacobi symbol, we have either U d ≡ 0 ( mod n ) {U_d} \equiv 0\,\pmod n or V d ⋅ 2 r ≡ 0 ( mod n ) {V_{d \cdot {2^r}}} \equiv 0\,\pmod n , for some r with 0 ⩽ r > s 0 \leqslant r > s . Let D = L − 4 Q > 0 D = L - 4Q > 0 . Then every arithmetic progression a x + b ax + b , where a, b are relatively prime integers, contains an infinite number of odd (composite) strong Lehmer pseudoprimes with parameters L, Q. Some new tests for primality are also given.

On the composite Lehmer numbers with prime indices, III

On the composite Lehmer numbers with prime indices, III

Corrigendum to the papers "On two theorems of Gelfond and some of their applications" and "On primitive prime factors of Lehmer numbers III"

Corrigendum to the papers "On two theorems of Gelfond and some of their applications" and "On primitive prime factors of Lehmer numbers III"

On primitive prime factors of Lehmer numbers III

On primitive prime factors of Lehmer numbers III

On primitive prime factors of Lehmer numbers II

On primitive prime factors of Lehmer numbers II