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# 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. Zbl 0156.04801.
- 2 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM. Springer, 1996. Zbl 0859.11003.

Related:

CauchyDavenportTheorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

11B75*no label found*

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