3 as a Ninth Power (mod p).
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
We provide a generalization of Scholz’s reciprocity law using the subfields and of , of degrees and over , respectively. The proof requires a particular choice of primitive element for over and is based upon the splitting of the cyclotomic polynomial over the subfields.
A generalization of the reciprocity law of multiple Dedekind sums
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
A necessary and sufficient condition for the primality of Fermat numbers
We examine primitive roots modulo the Fermat number . We show that an odd integer is a Fermat prime if and only if the set of primitive roots modulo is equal to the set of quadratic non-residues modulo . This result is extended to primitive roots modulo twice a Fermat number.
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 , with core and carry concepts for extension to integers.
All elite primes up to 250 billion.