This paper gives a simple, explicit counterexample to a conjecture of András Sárközy about sums and products modulo a prime. Sárközy had guessed that any subset A of the integers mod p that is slightly larger than half of the residues would generate every nonzero residue either as a sum of two elements of A or as a product of two elements of A. The author proves this is false: for every odd prime p≥5 there is a set A of size (p−1)/2 for which the residue 1 cannot be written as a sum of two elements of A nor as a product of two elements of A. Hence no positive margin above one half can force all nonzero residues to appear in those sum-or-product sets.

To explain the shapes involved, the paper works in F_p, the integers modulo p, and writes A+A for all sums a+a' with a,a' in A and AA for all products aa'. The union A* = (A+A) ∪ (AA) is the set of residues that A produces by these two operations. A short side result used here is elementary: if |A|>p/2 then A+A already equals all residues mod p, so the natural “threshold” where one can force everything to appear is exactly one half of the residues.

The counterexample is built by a graph trick. Make a graph whose vertices are all residues mod p and join two distinct vertices when their sum is 1 or their product is 1. Also put a loop on any vertex that by itself gives 1 as twice the vertex or as its square. A subset A avoids producing 1 by sum or product exactly when it is an independent set in this graph, meaning it contains no adjacent vertices and no looped vertex. The author classifies all connected pieces of this graph. There are two tiny exceptional components, possibly one extra two-vertex component when the polynomial x^2−x+1 has roots in F_p, and every other vertex sits in a six-cycle. These six-cycles come from the two involutions x↦1−x and x↦x−1 and form a classical six-term orbit related to the cross-ratio.