On the congruences $σ(n) ≡ a (mod n)$ and $n ≡ a (mod φ(n))$

On Congruences Mod

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...

Stratification mod n

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...

Pseudofinite counting functions mod n

An Euler characteristic χ is definable if for every definable function f : X → Y and every r ∈ R, the set {y ∈ Y : χ(f−1(y)) = r} is definable. If R = Z/nZ and M is a finite structure, there is a (unique) Euler characteristic χ : Def(M) → Z/nZ assigning every set its size mod n. This χ is always strong and ∅definable. If M is an ultraproduct of finite structures, then there is a canonical stron...

Universal Kummer Congruences Mod Prime Powers

DIRECTED GRAPHS DEFINED BY ARITHMETIC (MOD n)

Let a and n > 0 be integers, and define G(a, ri) to be the directed graph with vertex set V{0,1,..., n -1} such that there is an arc from x to y if and only if y = ax (mod ri). Recently, Ehrlich [1] studied these graphs in the special case a = 2 and n odd. He proved that if n is odd, then the number of cycles in G(2, ri) is odd or even according as 2 is or is not a quadratic residue mod n. The ...