# Erdős-Heilbronn conjecture

Let $A\subset{\mathbb{Z}}_{p}$ be a set of residues modulo $p$, and let $h$ be a positive integer, then

 $h^{\wedge}\!A=\{\,a_{1}+a_{2}+\cdots+a_{h}\mid a_{1},a_{2},\ldots,a_{h}\text{ % are distinct elements of }A\,\}$

has cardinality at least $\min(p,hk-h^{2}+1)$. This was conjectured by Erdős and Heilbronn in 1964[1]. The first proof was given by Dias da Silva and Hamidoune in 1994.

## References

• 1 Paul Erdős and Hans Heilbronn. On the addition of residue classes $\mod p$. Acta Arith., 9:149–159, 1964. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0156.04801Zbl 0156.04801.
• 2 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM. Springer, 1996. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003Zbl 0859.11003.
Title Erdős-Heilbronn conjecture ErdHosHeilbronnConjecture 2013-03-22 13:38:15 2013-03-22 13:38:15 bbukh (348) bbukh (348) 10 bbukh (348) Theorem msc 11B75 CauchyDavenportTheorem