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I think this answer from the Judaism site sums it up:
7 is the length of time for a natural cycle to transpire, often ending with holiness/sanctification of some sort (the pattern set by creation)
7 is special because it was set down by God as the natural cycle in creation.
In the same question above is this answer about other numbers:
It mentions the numbers:
1, (2?), 4, (5?), (6?), 7, 8, 10, 12, 40, and 70 all have significance.
See that answer for full details of their importance.
Also important is the number 50 for Jubilee (from this answer):
Toward the end of the book of Leviticus the 7 day measure that is most familiarly the Sabbath is unrolled across years to be the sabbatical year, and across decades to be the Jubilee. The Jubilee is the 50th year - the crown of the 7th 7.
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The word "seven" occurs 461 times in 390 verses in the NKJV.
The number seven is used many times throughout the Bible. It usually is significant of something being whole, complete or perfected. There are other numbers that are prevalent in the Bible as well. Someone above mentioned the number twelve, which usually signifies government (12 tribes, 12 apostles). Six is said to be the number of man, and three sixes the number of the beast. While it's all very interesting the truth of the matter is that Jesus died for our sins, ANYTHING else is insignificant in comparison.
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The number 7 has religious meaning in diverse religions, including Judaism, Christianity, Islam, Zoroastrianism, Hinduism and ancient Near Eastern paganism. Leon R. Kass says in The Beginning of Wisdom, page 52, the Mesopotamians (Babylonians and Assyrians), before the coming of the Bible, already reckoned seven-day cycles, connected with the phases of the moon. They set aside the seventh, fourteenth, twenty first, and twenty eighth days of the lunar months; they had their own Sabbath, sabattu or sapattu, the day of the full moon. It seems likely that the familiar Hebrew seven-day week is based on the Babylonian tradition, although going through certain adaptations. By the second century BCE, seven was also associated with the number of known planets. The concepts of a full week or the total number of planets can be associated with completeness.
Twenty and its multiples are other favoured numbers. In Noah's Flood, it rained for forty days and forty nights. In 'โThere Was No King in Israelโ: The Era of the Judges' (published in The Oxford History of the Biblical World, pages 139-140), Jo Ann Hackett points out that the number of years of peace brought about by each of the major judges, or the number of years of their ruling, is a multiple of 20, with the sole exception of Jephthah who ruled Israel for 6 years. Othniel's defeat of Cushan-rishathaim gave Israel 40 years of peace; after Ehud, 80 years of peace; after Deborah, 40 years; after Gideon, 40 years; then Samson is said to have judged Israel for 20 years. These numbers were no doubt fortuitous, as they helped the early tradents when passing the stories down orally before they were finally written down.
The number 17 is important in the numerology behind the Book of Genesis, especially in the ages of the Patriarchs:
Each lifespan involves a perfect square (5, 6, 7 in a numeric series) and the third factor also forms a series (7, 5, 3). For each patriarch the sum of the factors is 17.
Furthering the above formulas, Genesis says Abrahamโs wife Sarah lived to 127 years, which is the sum of these consecutive square numbers plus 17.