Heilbronnʹs conjecture on Waringʹs number (mod p) Original Research Article

Let p be a prime kp−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 image. We prove Heilbronnʹs conjecture that γ(k,p)much less-thank1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)less-than-or-equals, slantC(q)k1/q for phi(t)greater-or-equal, slantedq. A comparable lower bound is also given. We also establish exact values for γ(k,p) when phi(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.