UNIT FRACTIONS AND THE ERDÖS-STRAUS CONJECTURE

On a coloring conjecture about unit fractions

We prove an old conjecture of Erdős and Graham on sums of unit fractions: There exists a constant b > 0 such that if we r-color the integers in [2, br ], then there exists a monochromatic set S such that ∑ n∈S 1/n = 1.

Counting the Number of Solutions to the Erdős-straus Equation on Unit Fractions

For any positive integer n, let f(n) denote the number of solutions to the Diophantine equation 4 n = 1 x + 1 y + 1 z with x, y, z positive integers. The Erdős-Straus conjecture asserts that f(n) > 0 for every n > 2. To solve this conjecture, it suffices without loss of generality to consider the case when n is a prime p. In this paper we consider the question of bounding the sum ∑ p<N f(p) asy...

On the Erdös-Straus Conjecture: Properties of Solutions to its Underlying Diophantine Equation

SUMS OF k UNIT FRACTIONS

Erdős and Straus conjectured that for any positive integer n ≥ 2 the equation 4 n = 1 x + 1 y + 1 z has a solution in positive integers x, y, and z. Let m > k ≥ 3 and Em,k(N) = | {n ≤ N | m n = 1 t1 + . . .+ 1 tk has no solution with ti ∈ N} | . We show that parametric solutions can be used to find upper bounds on Em,k(N) where the number of parameters increases exponentially with k. This enabl...

Dwork's Conjecture 1 Dwork's Conjecture on Unit Root Zeta Functions