Heilbronn's conjecture on Waring's number (mod p)

Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)≪k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)⩽C(q)k1/q for ϕ(t)⩾q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ϕ(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a2+b2+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a2+b2=p.