Numerical properties of 7

Seven has many mathematical properties. Some of its more mundane properties are that it's the fourth prime number, the second Mersenne prime (2^3-1), and it's both a factorial prime and a primorial prime.

Seven is more interestingly both part of the only cousin prime triplet (3, 7, 11) and part of the only twin prime triplet (3, 5, 7) - see also 19 and 47.

It's also the smallest number of sides of a regular polygon not constructible with compass and straightedge - see 257 for why that is.

Additionally 7 is the smallest number to be an emirp (reversing the digits makes a different prime number) in any single base - see the entry for 17.

Seven is the smallest number with a reciprocal that has a repeating pattern of more than one digit: 1/7 = 0.142857 where the underlined digits repeat. It's the smallest number n with the property that its reciprocal's digits repeat with a period of n-1 - this leads to some interesting numbers like 142,857, whose first six multiples are rearranged versions of its digits.

Seven has interesting connections with another favorite number of mine, 17 - see the page for 17.

Testing for divisibility by seven is harder than it is for other nearby numbers. The best test is probably to subtract twice the last digit from the original number, and repeat until you get a number that you know is or isn't divisible by seven. For example, let's try 142,857:


14,285 - 2*7 = 14,271

1427 - 2*1 = 1425

142 - 2*5 = 132

13 - 2*2 = 9 - 9 isn't divisible by 7, so neither is 142,857.

Seven in our world

Seven is a number that has been given very much spiritual significance. Like with 3 and 5, a set of seven things allows all the elements in the set to be symmetrical with each other. Therefore, there are many famous 7s in the human world, and some of them include:

- the seven deadly sins

- the seven wonders of the ancient world

- 7 moving objects in the sky (sun, moon, Mercury, Venus, Mars, Jupiter, Saturn)

- seven days in a week

- seven ages of man (according to Shakespeare)

- the seven seas

- Snow White and the seven dwarves

- seven traditional continents (though I myself think it makes more sense to consider Europe and Asia as being one continent)

- seven books in the Harry Potter series

- seven dirty words you can never say on television - since that listing was made in 1972, it's now outdated since some words are more offensive now, and some now quite tame (e.g. piss and shit)

- zillions of other occurrences in religion (in fact, many more than any other number) and other places.

This shows that 7 often seems to be a magic number similarly to 3, with an inherent symmetrical appeal - for examples of famous 3's see my page on 3.

Another example of the cult significance of 7 is that many other cult numbers have seven as their last digit - 17, 27, 37, and 47 are all notable cult numbers, as is 7 itself. Cult numbers are closely related to psychologically random numbers, as numbers that end in seven are usually the most random-sounding (see the entries for 17 and 37 for more on that).

In addition 7 is known for its connotation of luck, further adding to its cult significance.

Seven is also notable as the smallest number that is usually not immediately recognizable with our number sense alone. Example:

o o

o o o

o o

You’ll definitely need to count for that. You may be thinking, “Isn’t there a better way to represent 7?” Yes, and here it is:

o o o o

o o o

This still allows us to recognize seven, but here we have a problem. Seven of something can be arranged in many different ways, and most of them don’t allow us to quickly recognize them. Five and six, though not usually immediately recognizable in a row, can generally be recognized in a group. Unfortunately, seven is beyond that and you usually will need to either count to recognize it or perceive it as three and four, or perhaps three and three with one in between.

You will need to bring out your secondary number sense, which allows you to accurately approximate numbers by looking at them. This secondary sense extends to about 50. To me, seven may be the smallest psychologically large number, as it’s the smallest number that really feels large as in hard to recognize. But then again, this is a very borderline case, seeing as seven things can be remembered as one block of brain memory quite easily (like the digits of 1,654,746).

The prefixes hepta- (Greek, such as heptagon) and sept- (Latin, such as septillion and the month September) are used for seven, and both are reasonably common, but far less common than those for five and six.


For reasons I can't quite pinpoint today, the number seven always had an appeal to me as a child, which grew further after I turned 7. It still seems to have an appeal to me today, and it remains my favorite of all numbers today. Strangely, I didn't put it on my "Very Important Numbers" list, probably because it didn't have properties that I considered cool - see 2, 17, 37, 69, and 96 for numbers that I did put in my Very Important Numbers list.

(pointless fact: even though I'm American, I write the number 7 like continental Europeans with a line through it, so it appears as 7, which I sometimes wonder why I even do but usually don't think about)

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