Let S k (m) := 1 k + 2 k + · · · + (m − 1) k denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for m ≥ 4 the ratio S k (m+1)/S k (m) of two consecutive power sums is never an integer. We will develop some techniques that allow one to exclude many integers ρ as a ratio and combine them to exclude the integers 3 ≤ ρ ≤ 1501 and, assuming a conjecture on irregular primes to be true, a set of density 1 of ratios ρ. To exclude a ratio ρ one has to show that the Erdős-Moser type equation (ρ − 1)S k (m) = m k has no non-trivial solutions.

Let N q be the number of solutions of the equation a 1 x 2 1 + · · · + a n x 2 n = bx 1 · · · x n over the finite field F q = F p s . L. Carlitz found formulas for N q for n = 3 and for n = 4 and q ≡ 3 (mod 4). He also expressed N q for n = 4 and q ≡ 1 (mod 4) in terms of Jacobsthal sums. In an earlier paper, we found formulas for N q when d = gcd(n − 2, (q − 1)/2) = 1 or 2. In this paper, we find formulas for N q when d = 4 and p ≡ 7 (mod 8); and when there exists an l such that p l ≡ −1 (mod 2d).

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Abstract. Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form x

We consider a diagonal equation, which can be reduced to the form [Formula: see text] over a finite field of characteristic p > 2. In 1997, Sun obtained the explicit formula for the number of solutions to an equation of this type when n is even. In this paper, we find explicit formulas for the number of solutions when n is odd, k = 2rh, and there exists a positive integer ℓ such that pℓ ≡ 2m-1h + 1 ( mod 2mh), m = 3 or 4, r ≥ m, h = 1 or 3.