The connection between Egypt and Israel is the most ancient "international relationship" of Israel. Almost 4000 years ago Israel was not yet the name of a nation but the joint name of 12 Hebrew tribes; those of the 12 sons of Jacob - or Yaakov - who was called also 'Israel' since his struggle with an angel in his dream. The Bible tells us that after years of drought Yaakov and his sons go to Egypt, where the Nile can feed all the hungry in the region. In fact they stay there for 400 years and multiply to a large nation whom Moses leads out of Egypt, to their promised land the Canaan, in one of humankind's best known story, the Exodus.
An even earlier connection with Egypt was Abraham's concubine, "Hagar the Egyptian", and his first son Ishmael born by Hagar. When Abraham's wife, Sarah, born Isaac, Ishmael lost his rights of the firstborn and Hagar had to go. Hagar is often considered to be an Arab woman - by mistake: she was of course ancient Egyptian and spoke that language, not Arabic. Ishmael is considered as the forefather of the Arab nation.
A due question here is whether the Bible can serve as a historic source. Archeological, historic, linguistic and other evidence suggest, it can. It is not less supported by facts than some of the news in todays' paper.
After the establishment of Israel in 1948 its most feared enemy was he great Egyipt. Three major wars were initiated by Egypt: in 1956 and in 1967 by Nasser and in 1973 the Yom Kippur War by President Anwar Sadat. The same President Sadat accepted Israeli PM Menahem Begin's invitation to Israel in 1977, leading to the Israel-Egyipt Peace treaty in 1979.
Relationships between the two countries saw ups and downs since then. After initial euphoric Israeli tourism to Egypt, today Israelis take a risk even by visiting Sinai, and Egyptian media and intellectuals show sometimes fierce hatred to Israel. However, the peace treaty holds firm, and Egypt is playing a positive role in promoting coexistence and peace in the region.
Egypt moderates and mediates sometimes positively in Palestinian-Israeli conflicts, particularly about Hamas-ruled Gaza's recent aggressions. Many believe that Gaza, fully left by Israel in 2005, hasn't yet shown maturity for becoming an independent state economically and socially, capable to live in peace with its neighbors. A growing autonomy under Egyptian auspices could probably help Palestinian Gaza to reach faster and safer an economic and social self-rule and maturity for peaceful co-existence.