In this article I develop the notion of the order of an element modulo n, and use it to prove the famous n 2 + 1 lemma as well as a generalization to arbitrary cyclotomic polynomials. References used in preparing this article are included in the last page. §1 Introduction I might as well state one of the main results of this article up front, so the following discussion seems a little more moti...

Let Σg be a closed, orientable, connected surface of genus g ≥ 1. The mapping class group Mod(Σg) is the group Homeo(Σg)/Homeo0(Σg) of isotopy classes of orientation-preserving homeomorphisms of Σg. It has been a recurring theme to compare the group Mod(Σg) and its action on the Teichmüller space T (Σg) to lattices in simple Lie groups and their actions on the associated symmetric spaces. Indee...

Recently Zuhair Abdul Ghafoor Al-Johar [12] has directed our attention to a syntactic constraint that is—on the face of it—tighter than NF’s device of stratification; in this little essay I consider a weakening, namely the generalisation of stratification to stratification mod n. So far the coterie of NFistes has considered neither the possibility that the class of unstratified formulæ in the l...

Given a prime p and cusp forms f1 and f2 on some Γ1(N) that are eigenforms outside Np and have coefficients in the ring of integers of some number field K, we consider the problem of deciding whether f1 and f2 have the same eigenvalues mod p (where p is a fixed prime of K over p) for Hecke operators Tl at all primes l ∤Np. When the weights of the forms are equal the problem is easily solved via...

Let n = p ν 1 1 · · · p νr r > 1 be an integer. An integer a is called regular (mod n) if there is an integer x such that a 2 x ≡ a (mod n). Let ̺(n) denote the number of regular integers a (mod n) such that 1 ≤ a ≤ n. Here ̺(n) = (φ(p ν 1 1) + 1) · · · (φ(p νr r) + 1), where φ(n) is the Euler function. In this paper we first summarize some basic properties of regular integers (mod n). Then in or...