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We provide a generalization of Scholz’s reciprocity law using the subfields $K_{2^{t - 1}}$ and $K_{2^{t}}$ of $ℚ (ζ_{p})$ , of degrees $2^{t - 1}$ and $2^{t}$ over $ℚ$ , respectively. The proof requires a particular choice of primitive element for $K_{2^{t}}$ over $K_{2^{t - 1}}$ and is based upon the splitting of the cyclotomic polynomial $Φ_{p} (x)$ over the subfields.

A generalization of the reciprocity law of multiple Dedekind sums

Masahiro Asano (2007)

Annales de l’institut Fourier

Various multiple Dedekind sums were introduced by B.C.Berndt, L.Carlitz, S.Egami, D.Zagier and A.Bayad.In this paper, noticing the Jacobi form in Bayad [4], the cotangent function in Zagier [23], Egami’s result on cotangent functions [14] and their reciprocity laws, we study a special case of the Jacobi forms in Bayad [4] and deduce a generalization of Egami’s result on cotangent functions and a generalization of Zagier’s result. Further, we consider their reciprocity laws.

A geometric proof of Kummer's reciprocity law for seventh powers

R. Clement Fernández, J. M. Echarri Hernández, E. J. Gómez Ayala (2011)

Acta Arithmetica

A necessary and sufficient condition for the primality of Fermat numbers

Michal Křížek, Lawrence Somer (2001)

Mathematica Bohemica

We examine primitive roots modulo the Fermat number $F_{m} = 2^{2^{m}} + 1$ . We show that an odd integer $n \geq 3$ is a Fermat prime if and only if the set of primitive roots modulo $n$ is equal to the set of quadratic non-residues modulo $n$ . This result is extended to primitive roots modulo twice a Fermat number.

A sharper bound for the least pair of consecutive k-th power non-residues of non-principal characters (mod p) of order k > 3

Richard Hudson (1976)

Acta Arithmetica

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A bound for the first k-1 consecutive k-th power non-residues (mod p)

A congruence involving the quotients of Euler and its applications (I)

A congruence involving the quotients of Euler and its applications (II)

A generalization of a result of Fermat.

A generalization of Scholz’s reciprocity law

A generalization of the reciprocity law of multiple Dedekind sums

A geometric proof of Kummer's reciprocity law for seventh powers

A necessary and sufficient condition for the primality of Fermat numbers

A new formulation of the law of octic reciprocity for primes =+-3 (mod 8) and its consequences.

A note on a paper by Brenner.

A Note on prime k-th power nonresidues.

A problem of D. H. Lehmer and its generalization (II)

A rational sixteenth power reciprocity law

A refinement of a theorem of Gerst on power residues

A remark on the criteria for3 to be a ninth power (mod p).

A sharper bound for the least pair of consecutive k-th power non-residues of non-principal characters (mod p) of order k > 3

A Theorem on Totally Mulitplicative Functions.

Additive structure of the group of units mod $p_{k}$ , with core and carry concepts for extension to integers.

All elite primes up to 250 billion.

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

Currently displaying 1 – 20 of 24